Talk:2610: Assigning Numbers - Revision history

172.70.126.221 at 09:31, 16 May 2022

2022-05-16T09:31:37Z

108.162.221.101 at 20:27, 4 May 2022

2022-05-04T20:27:33Z

172.69.33.115: /* Paradoxicality argument */

2022-04-29T18:09:59Z

‎Paradoxicality argument

🎄: Clean up now that I have my emoji edit summary (thank you While False)

2022-04-28T22:28:22Z

Clean up now that I have my emoji edit summary (thank you While False)

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172.70.130.161 at 21:27, 28 April 2022

2022-04-28T21:27:53Z

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172.70.214.81: /* Paradoxicality argument */

2022-04-28T21:26:15Z

‎Paradoxicality argument

172.70.214.81: /* Paradoxicality argument */

2022-04-28T21:20:49Z

‎Paradoxicality argument

172.70.214.81: /* Paradoxicality argument */

2022-04-28T21:19:12Z

‎Paradoxicality argument

172.69.33.83: /* Paradoxicality argument */

2022-04-28T20:06:43Z

‎Paradoxicality argument

172.69.33.83: /* Paradoxicality argument */

2022-04-28T20:04:17Z

‎Paradoxicality argument

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 :::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless,  was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)
 ::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC)
 I, for one, am very pleased with the current compromise. The use of ellipsis and the inclusion of "(ironically)" has totally sold me on it.  Also, if anyone knows how to make those notes where you have the little number you can click on to see the full explanation, I think the proof by contradiction part could benefit from having the parenthetical statements moved to notes.  I'm going to look up how to do it, and I'll try, but if it all goes horribly wrong...[[Special:Contributions/108.162.221.101|108.162.221.101]] 20:27, 4 May 2022 (UTC)

← Older revision		Revision as of 20:27, 4 May 2022
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	:::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless, was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase\|Underbase]] ([[User talk:Underbase\|talk]]) 10:43, 28 April 2022 (UTC)		:::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless, was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase\|Underbase]] ([[User talk:Underbase\|talk]]) 10:43, 28 April 2022 (UTC)
	::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83\|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81\|172.70.214.81]] 21:26, 28 April 2022 (UTC)		::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83\|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81\|172.70.214.81]] 21:26, 28 April 2022 (UTC)
		+
		+	I, for one, am very pleased with the current compromise. The use of ellipsis and the inclusion of "(ironically)" has totally sold me on it. Also, if anyone knows how to make those notes where you have the little number you can click on to see the full explanation, I think the proof by contradiction part could benefit from having the parenthetical statements moved to notes. I'm going to look up how to do it, and I'll try, but if it all goes horribly wrong...[[Special:Contributions/108.162.221.101\|108.162.221.101]] 20:27, 4 May 2022 (UTC)

@@ Line 82: / Line 82: @@
 ::I tried a cropped (and less controversial) version of my original statement, to see what you thought about it.--[[Special:Contributions/162.158.78.229|162.158.78.229]] 02:20, 28 April 2022 (UTC)
 :::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless,  was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)
-::::I won't argue wit that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC)
+::::I won't argue with that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC)

@@ Line 82: / Line 82: @@
 ::I tried a cropped (and less controversial) version of my original statement, to see what you thought about it.--[[Special:Contributions/162.158.78.229|162.158.78.229]] 02:20, 28 April 2022 (UTC)
 :::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless,  was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)
-::::I won't argue wit that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC)
+::::I won't argue wit that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC) -edited --[[Special:Contributions/172.70.214.81|172.70.214.81]] 21:26, 28 April 2022 (UTC)

@@ Line 82: / Line 82: @@
 ::I tried a cropped (and less controversial) version of my original statement, to see what you thought about it.--[[Special:Contributions/162.158.78.229|162.158.78.229]] 02:20, 28 April 2022 (UTC)
 :::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless,  was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)
-::::I won't argue wit that. (I'll also back off "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC)
+::::I won't argue wit that. (I'll also back off to "non-true non-false," since I'm unsure how to understand other definitions.). "Incompleteness" (rather than "inconsistency") is still the missing piece. One claim in the above explanation: "David Hilbert's famous proclamation "We must know, we will know" is simply incorrect," Ignores this qualification -- making it a misapplication of what Gödel actually proved. Maybe we can eventually know truth -- but the limited tools constituting Gödel's proof were simply not up to that task.--[[Special:Contributions/172.69.33.83|172.69.33.83]] 20:04, 28 April 2022 (UTC)

@@ Line 81: / Line 81: @@
 ::Censoring my opinion is not a legitimate "compromise." I recommend that you attempt to refute (or at least counter) my opinion instead. - Don --[[Special:Contributions/172.70.207.8|172.70.207.8]] 22:55, 27 April 2022 (UTC)
 ::I tried a cropped (and less controversial) version of my original statement, to see what you thought about it.--[[Special:Contributions/162.158.78.229|162.158.78.229]] 02:20, 28 April 2022 (UTC)
-:::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless, Gödel was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)
+:::I'm not entirely sure what you mean by "paradox"; to my knowledge, that word doesn't have a formal mathematical definition. I assume you mean a non-true non-false statement? (feel free to correct me) In which case, Gödel did not consider this because he was working within classical logic, wherein statements can either be "true" or "false" and there is no third value. The reason he chose classical logic is because mathematics is currently performed using classical logic. And although most proofs of "the Gödel sentence is true" are a bit wishy-woshy, you can actually formalise a proof within ZFC set theory (a theory based on classical logic) that the Gödel sentence is true for the standard natural numbers (see my comment above). Of course, you could reject ZFC (and base mathematics on something like [https://en.wikipedia.org/wiki/Paraconsistent_logic paraconsistent logic]) but you'll probably have a hard time convincing mathematicians. Regardless,  was more concerned with the incompleteness of the system than with the truth of the Gödel sentence, and doesn't mention truth at all in Theorem VI (the First Incompleteness Theorem) of his original paper.--[[User:Underbase|Underbase]] ([[User talk:Underbase|talk]]) 10:43, 28 April 2022 (UTC)