https://primo.ai/index.php?action=history&feed=atom&title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems 2026-04-17T18:09:19Z Revision history for this page on the wiki MediaWiki 1.31.0 https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=40152&oldid=prev 2026-01-05T20:20:07Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:20, 5 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]]&#160; ... [[Orch-OR]] ... [[TAME]] ... [[Protein Folding &amp; Discovery|Proteins]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <ins class="diffchange diffchange-inline">[[Life~Meaning]] ... </ins>[[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]]&#160; ... [[Orch-OR]] ... [[TAME]] ... [[Protein Folding &amp; Discovery|Proteins]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39958&oldid=prev 2026-01-03T17:00:54Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:00, 3 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <del class="diffchange diffchange-inline">[[Mindful Universe]] ... </del>[[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]] ... [[TAME]] ... [[<del class="diffchange diffchange-inline">Orch-OR</del>]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology<ins class="diffchange diffchange-inline">]]&#160; ... [[Orch-OR</ins>]] ... [[TAME]] ... [[<ins class="diffchange diffchange-inline">Protein Folding &amp; Discovery|Proteins</ins>]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39796&oldid=prev 2026-01-01T18:34:48Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 18:34, 1 January 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Mindful Universe]] ... [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]] ... [[<del class="diffchange diffchange-inline">Living Intelligence</del>]] ... [[Orch-OR]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Mindful Universe]] ... [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]] ... [[<ins class="diffchange diffchange-inline">TAME</ins>]] ... [[Orch-OR]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39525&oldid=prev 2025-12-29T22:11:21Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 22:11, 29 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Mindful Universe]] ... [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]] ... [[<del class="diffchange diffchange-inline">Morphogenesis</del>]] ... [[Orch-OR]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Mindful Universe]] ... [[Consciousness]] ... [[Loop#Feedback Loop - Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology]] ... [[<ins class="diffchange diffchange-inline">Living Intelligence</ins>]] ... [[Orch-OR]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39484&oldid=prev 2025-12-29T20:13:10Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:13, 29 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Consciousness]] ... [[<del class="diffchange diffchange-inline">Orch</del>-<del class="diffchange diffchange-inline">OR</del>]] ... [[<del class="diffchange diffchange-inline">Pleasure Principle</del>]] ... [[<del class="diffchange diffchange-inline">Loop#Feedback Loop </del>- <del class="diffchange diffchange-inline">Creating Consciousness|Feedback Loop - Creating Consciousness</del>]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <ins class="diffchange diffchange-inline">[[Mindful Universe]] ... </ins>[[Consciousness]] ... [[<ins class="diffchange diffchange-inline">Loop#Feedback Loop </ins>- <ins class="diffchange diffchange-inline">Creating Consciousness|Creating Consciousness]] ... [[Quantum#Quantum Biology|Quantum Biology</ins>]] ... [[<ins class="diffchange diffchange-inline">Morphogenesis</ins>]] ... [[<ins class="diffchange diffchange-inline">Orch</ins>-<ins class="diffchange diffchange-inline">OR</ins>]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39409&oldid=prev 2025-12-28T21:11:34Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 21:11, 28 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l20" >Line 20:</td> <td colspan="2" class="diff-lineno">Line 20:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[https://www.bing.com/news/search?q=Gödel’s+Incompleteness+Theorem&amp;qft=interval%3d%228%22 ...Bing News]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[Consciousness]] ... [[Orch-OR]] ... [[<del class="diffchange diffchange-inline">Cosmopsychism</del>]] ... [[Loop#Feedback Loop - Creating Consciousness|Feedback Loop - Creating Consciousness]]</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[Consciousness]] ... [[Orch-OR]] ... [[<ins class="diffchange diffchange-inline">Pleasure Principle</ins>]] ... [[Loop#Feedback Loop - Creating Consciousness|Feedback Loop - Creating Consciousness<ins class="diffchange diffchange-inline">]]</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* [[Time]] ... [[Time#Positioning, Navigation and Timing (PNT)|PNT]] ... [[Time#Global Positioning System (GPS)|GPS]] ... [[Causation vs. Correlation#Retrocausality| Retrocausality]] ... [[Quantum#Delayed Choice Quantum Eraser|Delayed Choice Quantum Eraser]] ... [[Quantum</ins>]]</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__NOTOC__</div></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39349&oldid=prev 2025-12-28T17:29:07Z

<p>‎<span dir="auto"><span class="autocomment">Liar&#039;s Paradox</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:29, 28 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l114" >Line 114:</td> <td colspan="2" class="diff-lineno">Line 114:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Connection (loosely) to Gödel ===</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Connection (loosely) to Gödel ===</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Gödel’s work is not the same as the Liar’s Paradox, but it uses a related move<del class="diffchange diffchange-inline">:</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Gödel’s work is not the same as the Liar’s Paradox, but it uses a related move<ins class="diffchange diffchange-inline">. </ins>Gödel constructed mathematical statements that “talk about” provability inside a formal system.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">* </del>Gödel constructed mathematical statements that “talk about” provability inside a formal system.</div></td><td colspan="2">&#160;</td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39348&oldid=prev 2025-12-28T17:26:52Z

<p>‎<span dir="auto"><span class="autocomment">Liar&#039;s Paradox</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:26, 28 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l81" >Line 81:</td> <td colspan="2" class="diff-lineno">Line 81:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Liar&#039;s Paradox =</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= Liar&#039;s Paradox =</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">== Liar’s Paradox ==</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">A </ins>classic logic puzzle that creates a contradiction using self-reference.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>&#160;</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">The &#039;&#039;&#039;Liar’s Paradox&#039;&#039;&#039; is a </del>classic logic puzzle that creates a contradiction using self-reference.</div></td><td colspan="2">&#160;</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most famous version is a sentence like:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most famous version is a sentence like:</div></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39347&oldid=prev 2025-12-28T17:25:48Z

<p>‎<span dir="auto"><span class="autocomment">Important caution (what Gödel does NOT automatically prove)</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:25, 28 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l71" >Line 71:</td> <td colspan="2" class="diff-lineno">Line 71:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;youtube&gt;gd6-12ClN14&lt;/youtube&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;youtube&gt;gd6-12ClN14&lt;/youtube&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">=</del>= Important caution (what Gödel does NOT automatically prove) <del class="diffchange diffchange-inline">=</del>=</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= Important caution (what Gödel does NOT automatically prove) =</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Gödel’s theorems do &#039;&#039;&#039;not&#039;&#039;&#039; automatically prove that:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Gödel’s theorems do &#039;&#039;&#039;not&#039;&#039;&#039; automatically prove that:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l79" >Line 79:</td> <td colspan="2" class="diff-lineno">Line 79:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Those are additional philosophical steps, and many mathematics</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Those are additional philosophical steps, and many mathematics</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">= Liar&#039;s Paradox =</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Liar’s Paradox ==</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The &#039;&#039;&#039;Liar’s Paradox&#039;&#039;&#039; is a classic logic puzzle that creates a contradiction using self-reference.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The most famous version is a sentence like:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">: &#039;&#039;&#039;“This statement is false.”&#039;&#039;&#039;</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">=== Why it’s a paradox ===</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Try to label the sentence as either &#039;&#039;&#039;true&#039;&#039;&#039; or &#039;&#039;&#039;false&#039;&#039;&#039;:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* If the statement is &#039;&#039;&#039;true&#039;&#039;&#039;, then what it says must be correct — so it is &#039;&#039;&#039;false&#039;&#039;&#039;.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* If the statement is &#039;&#039;&#039;false&#039;&#039;&#039;, then what it says is incorrect — which would make it &#039;&#039;&#039;true&#039;&#039;&#039;.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Either choice flips into the other. That’s the paradox.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">=== A simpler everyday version ===</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* “I am lying.”</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If the speaker is lying, then they’re telling the truth.&#160; </ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If they’re telling the truth, then they’re lying.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Same trap.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">=== What the paradox teaches ===</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The Liar’s Paradox shows that:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* &#039;&#039;&#039;self-reference&#039;&#039;&#039; can break ordinary true/false logic</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* some statements can’t be consistently assigned a truth value within a simple system</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">It’s one of the reasons logicians became careful about:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* how languages refer to themselves</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* what counts as a valid “statement” in a formal system</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">=== Connection (loosely) to Gödel ===</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Gödel’s work is not the same as the Liar’s Paradox, but it uses a related move:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Gödel constructed mathematical statements that “talk about” provability inside a formal system.</ins></div></td></tr> </table>

BPeat https://primo.ai/index.php?title=G%C3%B6del%E2%80%99s_Incompleteness_Theorems&diff=39346&oldid=prev 2025-12-28T17:16:02Z

<p>‎<span dir="auto"><span class="autocomment">The First Incompleteness Theorem (informal)</span></span></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 17:16, 28 December 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l39" >Line 39:</td> <td colspan="2" class="diff-lineno">Line 39:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;youtube&gt;I4pQbo5MQOs&lt;/youtube&gt;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;youtube&gt;I4pQbo5MQOs&lt;/youtube&gt;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>= The First Incompleteness Theorem <del class="diffchange diffchange-inline">(informal) </del>=</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>= The First Incompleteness Theorem =</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If a formal system is:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>If a formal system is:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr> </table>

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