Is infinity a necessary concept?

[javisot]

It would be interesting if a mathematical Finitist could come and explain their position in a clear and coherent way.

https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin. I'm doing a search on finitist or ultrafinitist mathematicians, but there are very few, obviously.

 

[pinball1970]

https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin. I'm doing a search on finitist or ultrafinitist mathematicians, but there are very few, obviously.

Leopold Kronecker? He hated Cantor's ideas

 

[WWGD]

Leopold Kronecker? He hated Cantor's ideas

Supposedly contributed to Cantor's breakdown

 

[Jaime Rudas]

Perhaps I am missing something, but I am not sure that every occurrence of infinity can be replaced by "without bound".

In this regard, I seem to recall that a 2-sphere is a finite but unbounded space

 

[PeroK]

In this regard, I seem to recall that a 2-sphere is a finite but unbounded space

That's a different usage of the term "unbounded".

 

[Roberto Pavani]

But I don’t think that it works for cardinality.

I agree that cardinality is a different issue.

Still, I am not entirely sure that my $\sin(1/x)$ example is completely unrelated to it.

Between $\varepsilon_N=\frac{1}{N}$ and $\varepsilon_{N+1}=\frac{1}{N+1}$, there are not only infinitely many real numbers; that interval has the same cardinality as $(0,1]$ itself.

That is one reason why I find it difficult to think of infinity only in terms of "without bound".

And the fact that, as one zooms toward $x=0$, the qualitative shape of $\sin(1/x)$ remains essentially the same is amazing to me. No matter how small the interval becomes, infinitely much structure remains.

Finally, taking off a mathematician's hat that I certainly do not deserve to wear (being only a humble engineer), I wonder whether a space such as $(0,1)$ can really be understood purely in terms of "amounts without bound".

Even between $1/N$ and $1/(N+1)$ there remain continuum many points.

This is one reason why I still struggle to see infinity as merely "without bound", though I may well be missing something.

 

[pinball1970]

Supposedly contributed to Cantor's breakdown

Yeah, what a sad story. Cantor shook up the idea of infinity from what I have read (lay person perspective) and the mathematics community said some negative stuff about his work at the time. Hilbert supported him at least. We discussed (you explained rather) the "set of all sets" issue previously on PF if you remember.

 

[WWGD]

Yeah, what a sad story. Cantor shook up the idea of infinity from what I have read (lay person perspective) and the mathematics community said some negative stuff about his work at the time. Hilbert supported him at least. We discussed (you explained rather) the "set of all sets" issue previously on PF if you remember.

So, from the perspective of this posts, sets, collections of sets , are unbounded, in the sense of not having a maximal set, i.e., no sets that contain any/every other set.

 

[Matterwave]

https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin. I'm doing a search on finitist or ultrafinitist mathematicians, but there are very few, obviously.

A little unfortunate that the mathematical section of the wiki is just some references and an anecdote someone else wrote trying to make fun of him.

 

[bhobba]

I agree. The analytic continuation of a series is only equal to the series in the region where the series converges.

I gave it my like because it highlights a tricky issue. The problem is that 'shenanicans', such as defining the sum of 'divergent' series by analytic continuation, actually have important applications in mathematical physics (eg series from perturbation theory). See the following lectures by Professor Carl Bendor (lecture 4, but the others include interesting stuff):
https://www.youtube.com/results?search_query=Mathematical+Physics+-+Carl+Bender

I like to think it's context-dependent.

Added later:
After reviewing the material, Carl thinks the issue is in the language we use - namely, the summation symbol. Really, it should,for divergent series, be something else to indicate we are trying to extract something that makes sense from something that, taking the summation symbol literally, does not make sense.

Thaks
Bill

 

[bhobba]

Yes, but again, that would place it firmly in the "worth the trouble" realm for me.

Yes.

Just as an aside, that is where all the trouble with Gödel's theorem comes from. Not so well known is that Tarski showed the real numbers (along with geometry) evade Gödel and are decidable.
https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers

By restricting things to the integers, things become somewhat tricky.

That said, it is hard to do away with the integers as they are more or less implied by the Zermelo-Fraenkel Axioms, which most mathematicians accept.


Thanks
Bill

 

[PeroK]

Just a technical point. The set of natural numbers is infinite, by definition. This is a question of cardinality. To say it is unbounded requires a definition of distance (a metric) on the set. The usual definition of distance is $n - m$, where $n \ge m$.

This may seem trivial, but if we consider a set of functions or matrices, these sets can be infinite without any immediate concept of the distance between two functions or two matrices. We can and do, of course, define various metrics on these more abstract sets and for a given metric, the sets may be bounded or unbounded.

Likewise, subsets of the rational or real numbers can be infinite and bounded. Or, infinite and unbounded. Although, they cannot be finite and unbounded.

Finally, whether a set is infinite is an inherent property of the set itself. Whether it is unbounded depends on the choice of metric. For example, if we define the distance between any two separate integers to be $1$, then this is a valid metric (check the definition). With this metric, the set of integers is bounded (as no two integers have a distance greater than $1$ between them.)

As this is a mathematics thread, it's important to understand these concepts.