My math got pretty rusty since college, so please forgive the naivete and imprecise formulation of my question. I vaguely recall a mathematician telling me at a party something to the effect that there is a solid mathematical theory that posits the existence of the maximum number and then draws interesting inferences from this premise. Is there any truth to this? If so, what are the key words by which to Google it?
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4You can define mathematical theories in any way you want, sure. Not sure if you are still talking about 'numbers', or what it means to be a 'maximal' number in the normal sense these notions are used in mathematics. So maybe call the domain 'schnumbers', come up with some axioms for those 'schnumbers', define what a 'maximal schnumber' is, and see if you can derive the existence of a maximal schnumber from those axioms. – Bram28 Dec 21 '20 at 00:03

2I think any radical thing like this would have to be following some very different definitions. If you follow Peano axioms, for example, with the typical order we define, we can prove no such maximum natural would exist very easily (I can post such a proof using these axioms if requested, but it's fairly selfexplanatory) – Riemann'sPointyNose Dec 21 '20 at 00:04

@Riemann'sPointyNose But it would be true that if you use the Peano axioms, and add the premise that there is a maximal number, you can draw some very interesting inference from that premise .... :) – Bram28 Dec 21 '20 at 00:06

1@Bram28 Well it would no longer be a consistent system (provided we use the typical order on the natural numbers), but actually wait I guess this is rather interesting still tbh. So yeah actually! It would be interesting – Riemann'sPointyNose Dec 21 '20 at 00:07

1@Riemann'sPointyNose Right ... so we can infer a contradiction, and from that, anything we want ... like 'pigs fly' ... that's what I meant by interesting inferences :) Anyway, we should be helping out the OP .... could we maybe remove some Peano axiom (the $\forall x \forall y (s(x) = s(y) \to x = y)$ would seem to be the obvious candidate) and make this work? – Bram28 Dec 21 '20 at 00:10

@Bram28 ah yeah removing injectivity should work! I guess the only problem is that the solution for ${\mathbb{N}}$ would no longer be unique, i.e. ${\{0,1\}}$ ${\{0,1,2\}}$ ${\{0,1,2,3\}}$... basically any finite list of consecutive natural numbers from $0$ would suffice as valid ${\mathbb{N}}$. Is this a problem? – Riemann'sPointyNose Dec 21 '20 at 00:16

7That mathematician might have been referring to [ultrafinitism](https://en.wikipedia.org/wiki/Ultrafinitism). But people say strange things at parties. – Robert Israel Dec 21 '20 at 00:21

@RobertIsrael Ultrafinitism, as I could get from a quick search, does seem to satisfy that person's claim, i.e., that we can postulate the existence of the largest number M. Thanks a lot!!! – user865086 Dec 21 '20 at 04:56

1"The philosophy is explained in Doron Zeilberger's article. Basically, it's the belief that there is a largest natural number!" https://math.stackexchange.com/questions/531/whatisultrafinitismandwhydopeoplebelieveit – user865086 Dec 21 '20 at 07:49

1"So I deny even the existence of the Peano axiom that every integer has a successor." https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf – user865086 Dec 21 '20 at 07:51

3I vote to reopen. From a logician's perspective this is an interesting question with extremely nontrivial answers. I wrote [earlier](https://chat.stackexchange.com/transcript/message/56539021#56539021) on a whole lot of mathematics related to finitism but some moderator deleted them for being too many... Suffice to say, there are ways to capture **something like** the existence of a maximum natural even though a literal "maximum natural" is quite meaningless. – user21820 Dec 24 '20 at 15:34

@Bram28: It occurs to me that you may be interested in looking at formal systems that are not as strong as PA and compatible with finitism, among other stuff that I mentioned in my linked comments above. =) – user21820 Dec 28 '20 at 19:10