HallsofIvy Posted July 2, 2020 Share Posted July 2, 2020 "I am a layman trying to understand above theorems. This could be a stupid question." There is no such thing as a stupid question! "Does these theorems imply that we actually cannot prove that 2+2 = 4???" Gosh, I may have to reconsider! No, Godel's theorem say that, given any set of axioms large enough to encompass the properties of the non-negative integers there must exist some theorem that can neither be prove nor disproved. It does not say that a specific theorem cannot be proved. In fact, if we were able to identify a specific theorem that can not be proved nor disproved, we can always extend the axioms, perhaps by adding that theorem itself as an axiom, so that theorem can be proved. Of course, there would then be still another theorem that cannot be proved nor disproved. 1 Link to comment Share on other sites More sharing options...

Strange Posted July 16, 2020 Share Posted July 16, 2020 This is a very readable explanation of Godel's proof: https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714 1 Link to comment Share on other sites More sharing options...

francis20520 Posted July 19, 2020 Author Share Posted July 19, 2020 On 7/17/2020 at 5:03 AM, Strange said: This is a very readable explanation of Godel's proof: https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714 "He also showed that no candidate set of axioms can ever prove its own consistency." So, does this mean that the Peano Axioms can never prove it's own consistency?? What does mean for simple arithmetic operations like addition and multiplication??? Because consistency means uniformity or reliability. So does this say that we might sometimes get different answers to 2 + 3??? What does it mean??? Link to comment Share on other sites More sharing options...

Strange Posted July 20, 2020 Share Posted July 20, 2020 On 7/19/2020 at 6:14 PM, francis20520 said: So, does this mean that the Peano Axioms can never prove it's own consistency?? Correct. On 7/19/2020 at 6:14 PM, francis20520 said: What does mean for simple arithmetic operations like addition and multiplication??? Nothing. It means you could contrive a statement that can be written using the Peano axioms, but could not be proved to be true or false, using those same axioms. (That statement would be useless, other than as an example of Godel's theorem.) On 7/19/2020 at 6:14 PM, francis20520 said: Because consistency means uniformity or reliability. Not in this case it doesn't. Here it means: every statement can be proved to be either true or false. On 7/19/2020 at 6:14 PM, francis20520 said: So does this say that we might sometimes get different answers to 2 + 3??? No. Link to comment Share on other sites More sharing options...

uncool Posted July 21, 2020 Share Posted July 21, 2020 1 hour ago, Strange said: Not in this case it doesn't. Here it means: every statement can be proved to be either true or false. You're thinking of completeness. Consistency means that no statement can be proven both true and false. 1 Link to comment Share on other sites More sharing options...

Strange Posted July 21, 2020 Share Posted July 21, 2020 7 hours ago, uncool said: You're thinking of completeness. Consistency means that no statement can be proven both true and false. Thank you Link to comment Share on other sites More sharing options...

HallsofIvy Posted July 31, 2020 Share Posted July 31, 2020 On 7/19/2020 at 12:14 PM, francis20520 said: "He also showed that no candidate set of axioms can ever prove its own consistency." So, does this mean that the Peano Axioms can never prove it's own consistency?? What does mean for simple arithmetic operations like addition and multiplication??? Because consistency means uniformity or reliability. So does this say that we might sometimes get different answers to 2 + 3??? What does it mean??? That's not what "consistency" means. Saying the Peano axiom are consistent means that the Peano axioms cannot be used to prove both statement "p" and "not p". Saying that the Peano axioms cannot be used to prove its own consistency means that the axioms cannot be used to prove that statement. Link to comment Share on other sites More sharing options...

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