Timeline for How do you prove mathematical induction without the notion of a set?
Current License: CC BY-SA 4.0
5 events
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| Jun 21 at 2:58 | comment | added | Corbin |
@leepappas: From this Math SE discussion, there's a model of the Peano axioms which doesn't satisfy induction. (Several of them, in fact!) If the other axioms proved induction, then all models would have it. I love the polynomial ring example: In N[x], the number x is neither even nor odd.
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| Jun 20 at 14:49 | comment | added | mudskipper | @leepappas - Even though ChatGPT cannot be trusted in general -- if you ask ChatGPT (free version) to clarify (1) what is meant by the least inductive set in ZF and (2) what is the relation between the concept of least inductive set and the principle of induction, it will also happen to give correct answers that may clarify your questions. | |
| Jun 19 at 8:29 | comment | added | Mikhail Katz | en.wikipedia.org/wiki/Robinson_arithmetic is an example of a theory that incorporates the other axioms (successor, etc.) but does not include the induction schema. It is known to be a weaker theory of arithmetic. | |
| Jun 18 at 22:51 | comment | added | lee pappas | How do you prove that you can't prove it, knowing the other axioms? My question assumes you can prove it. I don't want to use ZF, or any set theory for that matter. I want to use FOL treating "is a natural number" as a predicate. | |
| Jun 18 at 11:10 | history | answered | Mikhail Katz | CC BY-SA 4.0 |