Is infinity a necessary concept?

[dendros]

Well, that is not my objection but of those who reject infinity in math.
They seem to conflate our limits with math's limits.

 

[Mark44]

But to negate the infinity of the set of naturals, for example, is like negating math since addition with 1 guarantees that a number can always be increased so there is not a last natural number.

Now, some might say that you cannot actually perform this addition an infinite number of times, which is true.
This does not negate the actual infinity of the set of naturals but rather expresses that we have finite resources.

Did you instead mean to say "dispute the infinity of ..." rather than "negate"?

 

[dendros]

Well, finitism by definition is the negation of infinity.

 

[Matterwave]

No. I think it is part of the "Big Bang" expansion theory. But I will not say more, because I do not know more. I'll leave it to others.

Somewhat off topic to the thread, but just to close the loop on this: in our cosmological models the *observable* universe is finite due to the finite speed of light and the age of the universe being finite (~13.8Gyr). "The universe" has unknown status. Einstein's eqns admit both spatially finite (closed is the better term, there's no edge but the universe loops back in on itself) and infinite solutions. Our most basic data points to a roughly flat FLRW metric which would correspond to an infinite universe.

Note that even if the universe is finite in size/closed, we still model it using a continuous mathematical object (manifolds) on which we can do calculus (take limits, differentiate, etc.). So infinity still pops up in the sense that the continuum between (0,1) is infinite.

Indeed, it seems to me like Finitists have a lot of theory to develop if they want their mathematics to reach anywhere near the utility of standard mathematics.

 

[jeffn1]

When applied to the physical universe, it seems to me the concept requires some sense of continuity of the "thing" considered infinite. For example up until the '60s, we thought our Milky Way Galaxy was "the universe". And people argued that it was "infinite". Then we realized the we inhabit a particular galaxy and there are billions of other galaxies. It is likely that things morph into different structures at the largest and smallest scales. It seems unlikely that that concept of infinite properly applies to the physical world. It seems to me to be a "placeholder" or conceptual/mathematical tool. (Sorry, a bit off topic)

 

[Dale]

it seems to me like Finitists have a lot of theory to develop if they want their mathematics to reach anywhere near the utility of standard mathematics

Yeah, I mean you cannot even use the typical axioms of addition on the natural numbers.

 

[Jaime Rudas]

For example up until the '60s, we thought our Milky Way Galaxy was "the universe". And people argued that it was "infinite".

Actually, no. As early as the mid-18th century, Immanuel Kant and Thomas Wright proposed the "island universes" hypothesis—the idea that nebulae were, in fact, collections of stars similar to the Milky Way. This idea captivated a significant number of scientists, as illustrated by the famous Great Debate between Shapley and Curtis, which culminated in 1924 when Edwin Hubble demonstrated that Andromeda was, in reality, another galaxy.

 

[WWGD]

-1/12 is the value of the analytic continuation of the Riemann Zeta ( $\Sigma_n=1 ^ {\infty) \frac {1}{n^s }$ , at s=-1. Have @mathwonk or @fresh_42 verify it. The continuation doesn't have anice closed form, else the Riemann Hypothesis could be easily answered.

 

[Matterwave]

-1/12 is the value of the analytic continuation of the Riemann Zeta ( $\Sigma_n=1 ^ {\infty) \frac {1}{n^s }$ , at s=-1. Have @mathwonk or @fresh_42 verify it. The continuation doesn't have anice closed form, else the Riemann Hypothesis could be easily answered.

I covered this in posts 13 and 18.

 

[WWGD]

We have very powerful mathematics thanks to dealing with the finite and the infinite; it would be less powerful if we only treated the finite as a mathematical object. Fewer tools, less depth, less rigor, etc.


(I have a question: in a context of "only finite mathematics" are incompleteness theorems still relevant? I would say no, but I prefer to ask.)

I think no, if you mean a finite universe of elements. Not sure if you need finitely- many predicates too. Then just checking finitely many conditions is your finite decision process to determine the truth.

 

[zzephod]

In understanding the truth of things, is it important that we can not perform the addition an infinite number of times? What if God can do it?
(I hesitate to put it that way and possibly sidetrack the discussion, but I don't see how the physical limitations of humans should limit our conceptual understanding. If a human could perform each addition in half the time of the prior addition, he could complete it quickly.)

Remind me... which axiom set includes this "God" concept.

What you want is an appropriate theory/model of hyper-computation, which allows the completion of a countable infinity of operations in finite time... good-luck building a processor that realises such a model in this universe.