I would like to re-address the issue of the OP, expressed in the title:
(What is the) The Largest Number We Will Ever Need (?)
I'd say one, if all other axioms in the mainstream system are applied. 1. It can be added, the sums can be multiplied or divided or subtracted from each other, and the whole shkebam can be developed just form one number, which is one.

This is a specific example of the largest number. It could be a half, a million, any number, really, real or imaginary, and rational or irrational. Any one number could satisfy the question, "What is the largest number we shall ever need?" Provided, of course, that the generation of other values, expressed in numbers, in an infinite variety, is possible from the axioms used.

That's wonderful. :up:

All we need to get the ball rolling is 1.

Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. — Charles Darwin

Le Monad.

Which means ∞∞ is impossible, squaring with Aristotle's decision to make the distinction potential vs. actual (infinity). — Agent Smith

Can infinity even be considered as possible or impossible? it's a concept. If an axiom calls it to be true/in need then I see no reason to assume otherwise.

Can infinity even be considered as possible or impossible? it's a concept. If an axiom calls it to be true/in need then I see no reason to assume otherwise. — john27

I haven't read the original argument made by Aristotle - Wikipedia offers only a rough sketch. It seems as though Aristotle considered real/actual entities as those that had an end; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. The same goes for $\infty$, it, by definition is endless.

I haven't read the original argument made by Aristotle - Wikipedia offers only a rough sketch. It seems as though Aristotle considered real/actual entities as those that had an end; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. The same goes for ∞∞, it, by definition is endless. — Agent Smith

I don't agree that it's a valid application of Aristotle's rule of the difference between actual and potential.

The infinity is there. The only reason it can't be counted is because counting is a process which is always finite. However, it is the tool of the test, the counting, that is the culprit here, so to speak; it is the weakness of the tool that stops us from realizing the actuality of infinity.

If there were an instrument that meausred infinity, then the actuality would immediately show through.

If there were an instrument that meausred infinity, then the actuality would immediately show through. — god must be atheist

I think this is a good point. It would be harmful to assume that infinities don't exist in the real world simply because we can't process them in a tangible manner.

Yeah, we need a different tool!

d; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. — Agent Smith

What about a cyclical process?

What about a cyclical process? — john27

I proposed that; no takers!

Why? First Cause?

If, a big if, there did exist a finite number Nmax that could stand in for, salva veritate, ∞∞, we could prove/disprove all mathematical conjectures via proof by exhaustion (brute search) with the help of existing supercomputers — Agent Smith

Some mathematical systems require an infinite set to function. I think it's just a question of where your looking at; assuming infinity isn't intrinsically false, or bad, it's just another way to look at something, which happens to have some pretty interesting conclusions.

Here's a page I found that I think is relevant to the topic:

https://en.wikipedia.org/wiki/Axiom_of_choice

Why? First Cause? — john27

Dunno!

"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

I will say that the benefit of virtue ethics is that you'll no longer have to reconsider this in straightforward situations that are not ethical dilemmas. In acting virtuously, virtuous action becomes habit

"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

That makes it appear that I said that we can't expect that posts have good outcomes, etc.

But what I posted:

Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

That is to say that I don't expect that ordinarily posters ask those questions before posting.

In acting virtuously, virtuous action becomes habit — Kuro

I understand that view.

Total possible universes = every possible arrangement of Plank spaces and Plank times?

Well to begin with i would calculate all the countable space in the universe in the smallest units possible (Planck lengths). Then i would calculate the total lifespan of the universe in Planck time units. After that i multiply the last two results to get the total amount of countable space in the universe times the amount of countable time in the whole lifespan of the universe.

Total space in universe calculation:
------------------------------------
Planck length in meters:
1.6 x 10^-35 =
0.000000000000000000000000000000000016

Diameter of the universe in meters:
8.8 x 10^26 =
880000000000000000000000000

Diameter of universe in Planck lengths:
5.4453761 x 10^61 =
54453761000000000000000000000000000000000000000000000000000000

Volume of universe in Planck cubes:
1.61467 x 10^185 =
161467000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


Total time of universe calculation:
-----------------------------------

Current age of universe till now in years:
13.7 x 10^9 =
13700000000

Time from now till universe heat death in years:
1.7 x 10^106 =
17000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Total life (past + future) of universe in years: (insignificant difference)
1.7 x 10^106 =
17000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Total life of universe in seconds:
5.3618 x 10^113 =
536180000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

A few more calculations...

Planck lengths per light second (number of plank lengths light travels in 1 second):
1.855 x 10^43 =
18550000000000000000000000000000000000000000

Universe Planck volumes per second over total life of universe:
9.946139 x 10^156 =
9946139000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

Universe Planck volume * Planck time instances over life of universe:
1.605973225913 x 10^342 =
1605973225913000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


After that i can add matter into the mix by calculating the number of subatomic and atomic particles in the universe at any one plank time instance. Then we multiply that number into our last result.


Number of subatomic and atomic particles in the universe:
3.28 x 10^80 =
328000000000000000000000000000000000000000000000000000000000000000000000000000000

Result for all countable things in the universe so far:
5.267592180994640 x 10^422 =
526759218099464000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


We can add a few more things like stars planets and galaxies:
-------------------------------------------------------------
Stars in the universe:
10^24 =
1000000000000000000000000

number of planets in the universe (star orbiting and rogue planets):
10^51=
1000000000000000000000000000000000000000000000000000

number of galaxies in the universe:
13 x 10^15 =
13000000000000000

All stars, planets, and galaxies in the universe:
1.3 x 10^91 =
13000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

My final result:
6.847869835293032 x 10^513 =
6847869835293032000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


This number can be significantly larger if we multiply in the number of molecules in the universe along with the number of space dust particles, asteroids, comets, black holes, and whatever else i'm missing.

Surely you jest.

I jest you not!
And don't call me Surely.

Ta Da !!! Now, officially, there are new Biggest and Smallest Named Numbers.

Here comes . . . . . . Zepto !

:nerd:

Question: What's the Nmax for our universe? — Agent Smith

Very easy: aleph-1. The the infinite cardinal of the real numbers

Because I think there's still something for us to understand with infinity, it isn't so easy that to use finite logic. And more interestingly, bigger infinities seem not to be usefull for example in physics, computing, etc.

aleph-1. The the infinite cardinal of the real numbers — ssu

card(reals) = aleph_1 is the continuum hypothesis. It is not provable in ZFC. It is thought to be true by some mathematicians and false by other mathematicians - an unsettled question.

- an unsettled question. — TonesInDeepFreeze

What did I say? There's still something for us to understand with infinity.

Actually, I think the continuum hypothesis is that from aleph_0 the next is aleph_1.

Actually, I think the continuum hypothesis is that from aleph_0 the next is aleph_1. — ssu

It is correct that the next aleph after aleph_0 is aleph 1. That follows trivially from the definition of the alephs. Since the alephs are indexed by the ordinals, and 1 is the next ordinal after 0, It is trivially the case that the next aleph after aleph_0 is aleph_1, regardless the continuum hypothesis. Meanwhile, actually, it is incorrect that "the next aleph after aleph_0 is aleph 1" is the continuum hypothesis.

The continuum hypothesis is that the cardinality of the set of real numbers is aleph_1. Or, equivalently, that 2^aleph_0 = alpeh_1.

What did I say? There's still something for us to understand with infinity. — ssu

You said there's more for us to understand regarding infinitude. Indeed, whether the cardinality of the set of real numbers is aleph_1 is something more that is not clearly understood. Again, settling that question is to settle the continuum hypothesis.

But that's about infinity, actually. The Continuum Hypothesis.

card(reals) = aleph_1 is the continuum hypothesis. It is not provable in ZFC. It is thought to be true by some mathematicians and false by other mathematicians - an unsettled question. — TonesInDeepFreeze

It's unsettled because there's a problem with what constitutes a "countable" cardinality. As soon as we define "countable" such that an infinite set might be "countable", we create incoherency.

Yes, the continuum hypothesis is about the first two infinite cardinals. Meanwhile, what I said stands:

aleph-1. The the infinite cardinal of the real numbers — ssu

That is the continuum hypothesis.

In other words,

The cardinality of the set of real numbers is aleph_1

is the continuum hypothesis.

I think the continuum hypothesis is that from aleph_0 the next is aleph_1. — ssu

That is not the continuum hypothesis.

In other words

aleph_1 is the next aleph after aleph_0

is not the continuum hypothesis.

As to the other poster, the current question of the continuum hypothesis does not stem from the definition of 'countable'. Rather, the current question stems from the fact that the continuum hypothesis is independent of ZFC.

aleph_1 is the next aleph after aleph_0

is not the continuum hypothesis. — TonesInDeepFreeze

The cardinality of the set of real numbers is aleph_1

is the continuum hypothesis. — TonesInDeepFreeze

As to the other poster, the current question of the continuum hypothesis does not stem from the definition of 'countable'. — TonesInDeepFreeze

Well, it's about what can be put into one-to-one correspondence with the set of natural numbers and the reductio ad absurdum proof that this cannot be done with the set of reals. Here 'countable' has it's problems, when ordinarily everything that we can map into one-to-one correspondence is countable (a+b=c).

When ZF was meant to do away with the paradoxes, it's obvious that it has problems with infinity. After all, it's just taken as an useful axiom.

It seems that you wish not to recognize that:

aleph-1. The the infinite cardinal of the real numbers — ssu

is the continuum hypothesis

and that

the continuum hypothesis is that from aleph_0 the next is aleph_1 — ssu

is incorrect, since "from aleph_0 the next aleph is alelp_1" is not the continuum hypothesis.

/

what can be put into one-to-one correspondence with the set of natural numbers and the reductio ad absurdum proof that this cannot be done with the set of reals. — ssu

Cantor's proof was not by reductio ad absurdum.