You would claim that the set of natural numbers has a fixed size?

If you cannot count something, how can it have a size?

Your mistake (and Cantor's) is thinking that 'larger' could make sense in terms of infinite sets.

Something infinite has no fixed size so it cannot be compared to anything else. An infinite set has no size (cardinality) is the only valid conclusion and thus they cannot be compared.

What Cantor did (in his madness) was to invent numbers to represent the sizes of infinite sets... they are pure inventions... derived from nothing... there is no math behind them... its all a pure figment of Cantor's imagination.

But for any finite interval, we find more members in the second set than the first set. So we inductively understand that the second set is 'larger' that the first set.

We certainly can't conclude that they are equal (as set theory does) as they are not fully defined as indicated by the ellipses.

But bijection claims that these are of the same size:

{ 1, 4, 9, 16, ... }
{ 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, ... }

When the 2nd is clearly larger than the first. So how exactly is that a sound resolution to the paradox?

How for example has Galileo's paradox been resolved?

'Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work, Two New Sciences, Galileo Galilei made apparently contradictory statements about the positive integers. First, some numbers are squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets.

Galileo concluded that the ideas of less, equal, and greater apply to (what we would now call) finite sets, but not to infinite sets. In the nineteenth century Cantor found a framework in which this restriction is not necessary; it is possible to define comparisons amongst infinite sets in a meaningful way (by which definition the two sets, integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than others.'

https://en.wikipedia.org/wiki/Galileo%27s_paradox

It remains the case that bijection claims that the number of non-squares is the same as the number of squares. But in every finite interval (of reasonable size) we examine, the number of non-squares is greater than the number of squares. The paradox still stands. The resolution to the paradox is not Cantor's bijection procedure but Galileo's earlier observation that infinite sets cannot be compared. An infinite set is not fully defined (that is what the ... indicates), meaning it is not defined period and it is not permissible to perform operations on it (like comparing sizes).

If you could kindly point out where these have been shown to be faulty?

I assure you I would not repeat an argument if it had a known fault in it.

No, you haven't reasonably justified the entirely assumed necessity of a first cause. — S

If there was no first cause, there would be an infinite regress in time. That is impossible; an infinite regress has no start; the start defines the first event, the first event defines the 2nd, and so on - so we can conclude that none of the events in a infinite regress is defined; an infinite regress is impossible.

Plus I have argued that there is a start of time and that clearly requires a timeless first cause.

Also, something has existed always; the only way to exist always is outside of time; to have no cause and thus be the first cause.

I've shown your argument presented here to be faulty and you haven't resolved the fault. — S

I do not agree that you have shown a fault with any of the three arguments.

For a start, these two statements contradict one another. You clearly do know of objections to your argument. Presumably Cantor, and those who follow him, haven't just written "some infinities are bigger than others" on the back of an envelope and that's what's guided mathematics for the last hundred years. — Isaac

That is what they have done. The axiom of infinity could fit on the back of an envelope and just baldly states that infinity exists. They have no logical justification for including Actual Infinity in maths.

Lay out the arguments Cantor, and others, have made, and show exactly where they went wrong. That way people here (probably not me) can actually get involved in the debate. — Isaac

For example the bijection procedure gives spurious results (like naturals and rationals being the same size). Or all the paradoxes listed here:

https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals

A paradox is usually the sign that you are working with something contradictory. The common thread through all these paradoxes is infinity - it is the contradictory thing that leads to those paradoxes.

Don't assume that it is. — S

Well the 'something' must logically contain the first cause. The first cause is the explanation for everything else so my substitution is valid.

You must not know what that means. — S

I know precisely what that means.

An uncaused, timeless, first cause.

I have given 3 valid arguments for this position.

At last some sense... eternalism is possible.

It seems to me that the members of this site are mostly presentists.

I am often outnumbered when debating the issue. Maybe we should have a poll of presentism Vs eternalism?

Because they mean different things. If I were to say that, "I'm going home", and, "I'm going fishing", are equivalent, then I'd be talking rubbish. — S

If the 'something' in 'why is there something rather than nothing?' is the first cause then we have the question:

'why is there a first cause rather than nothing?'

There is nothing logically prior to the first cause. So the answer to 'why is there something rather than nothing?' in 'no reason'.

If there is nothing logically prior to the first cause; there is no possibility of any explanation except that given above.

No, that's not "i.e. the first cause". That's completely unreasonable. — S

If there ever was a state of nothingness in the past; there would be nothing now (because you can't get something from nothing).

So something must have always existed. IE A first cause.

I wasn't asking for a repeat of your assertion, I was asking you to address the counter arguments of mathematicians — Isaac

But I know of no objections to my argument. Hence my reason for posting this argument is to gather any such objections. Philosophers tend to be more open minded and have less invested in infinity than mathematicians, hence this audience.

I have a 1st in maths so I do understand the maths.

It is the case that there is a huge flaw in the root of set theory called 'the axiom of infinity'. As I mentioned to you before, there was a big debate at the turn of the century as to whether infinity should be included in maths. Cantor won the debate and its been taught in schools ever since.

So I think there is groupthink going on here. There are only a few people around like me who are actually speaking out on the obvious flaws of infinity. I also think mathematicians are working at too high a level of abstraction - the flaws are obvious in that, for example, infinity is completely incompatible with basic arithmetic - but most mathematicians cannot see the wood for the trees - they are too immersed in higher level abstractions to see the basic problems.

I believe it goes without saying that either you have an effect without a cause or that something existed forever — christian2017

I have an argument that rules out the 2nd:

https://thephilosophyforum.com/discussion/5302/an-argument-for-eternalism/p1

So I think therefore there must be an effect without a cause; something beyond causality, IE beyond time that is the first cause of everything.

Well reasoned counter argument :(

No they're not. — S

Please say why are they not equivalent.

You aren't justified in suggesting that there's a first cause. That's an act of faith. — S

There are different ways to show there is a first cause:

1. The argument in the op: can't get something from nothing so something (IE the first cause) must have existed always.

2. If you think about time stretching back; it forms an infinite regress. Maybe if we had another time, say time2 and that created time? Then we'd need a time3. So time always results in an infinite regress. The only way out of this infinite regress is something timeless. IE a timeless first cause.

3. I believe time has a start (see https://thephilosophyforum.com/discussion/5302/an-argument-for-eternalism/p1) so that also requires a timeless first cause

The circumference of the topographical position of any country, is infinite. — SethRy

Eventually, zooming in, you reach atom level. So the circumference is not infinite.

You haven't addressed the arguments of those mathematicians who advance modern infinity theory, you've just declared them all to be 'wrong'. — Isaac

I have addressed them - sorry for repeating myself- but If infinity were a number it would be a number X such that it is greater than all other numbers. But X+1>X so infinity is not a number. If you can disprove this argument please tell me how - I've been posting it for months and no-one seems to have a valid counter argument.

If you look at set theory, all it does is axiomatically declare that an infinite set exists - it does not prove anything at all and yet its easy to show infinity does not exist - see above - or see any of the numerous contradictions that are thrown up with infinity.

- There are proofs that infinity does not exist.
- There are no proofs that infinity does exist.
- What exactly can we deduce from that apart from infinity does not exist?

Modern physics seems to be leaning in the direction of spacetime having a start:

https://www.michaelgstrauss.com/2017/01/the-significance-of-bgv-theorem.html

The BGV theorem states, as you'd expect, that an expanding universe must have a start.

If we all held the same viewpoint there would be nothing to discuss thus no philosophy discussion forum.

On the subject of idiosyncratic definitions of infinite:

∞+1=∞

In english, this means there exists something that when you change it, it does not change. Absolute nonsense.

The questions "why is there something rather than nothing" and "why is this something the way it is" both are equivalent to asking "What is the explanation for the First Cause?".

But the First Cause can have no explanation; there is no cause of the first cause; no reason for it. The first cause has to be timeless and thus beyond causation (else we end up in an infinite regress).

The physics of the situation that is wrong with an infinite universe:

1. An infinite universe should have no boundaries so cannot expand

2. There is nothing beyond an infinite universe to expand into

3. Expanding means that size(t1) > size(t0). IE we know the universe was once smaller than it is today. There is only one kind of infinity here; size is an Aleph One infinity. So size(t0) must be a finite number.

4. The universe can't have been expanding for ever; if we trace back in time far enough, we will always find a point in time when two distant points are co-located so no further expansion is possible. Hence the universe must have finite size.

The 'size' of the universe" MAY not be a number at all. — Frank Apisa

Thats a contradictory statement - size is always a number.

Your pontifications are used gratuitously in order for you to arrive at "the universe is not infinite" which you need for wherever you ultimately want to go.

Can you truly not see that? — Frank Apisa

If we adopted your approach to philosophy and science which seems to consist of 'I don't know' and 'its unknowable' I think progress would grind to a halt.

Whether space is expanding along with the galaxies or it is constant while the galaxies expand within it is yet to be determined. — BrianW

Galaxies are flying apart at faster than the speed of light so it is space that is expanding I believe.

The universe could just be finite in time and space. That is a nice simple model that fits the facts.

I don't think it's the case that we need some 'number' other than infinity to describe the universe. The size of the universe is a number, numbers have fixed values, so the size of the universe is a finite number.

Trying to clarify my last remarks:

If infinity is a number; it is a number larger than any other number. So if infinity+1 (IE it grows) is a number, then infinity is not infinity.

If infinity is a number; it is a number larger than any other number. There can be only one such 'number larger than any other number'.

The dictionary definition of infinity is:

'MATHEMATICS
a number greater than any assignable quantity or countable number (symbol ∞)'

So if infinity grows, it was not infinity to start with (because there was a number greater than infinity).

By the same definition, it is not possible to have more than one infinity, although as you say maths goes on to contradict its own definition of infinity by introducing multiple infinities.

Cantor imagined a hierarchy of different sized infinities terminating with 'Absolute Infinity' at the top. This concept Cantor said was beyond maths and he associated it with God. Cantor believed God communicated these ideas to him directly. Cantor was a looney IMO.

Distant galaxies are flying apart at a rate that greater than the speed of light so it appears it is space is expanding.

I guess it is expanding within nothingness. So the nothingness has no space or no time associated with it. Maybe time slows down as you get closer to the edge of the universe and stops on the boundary? If the boundary was expanding at the speed of light, that would make some sense.

Fair point. Sorry I have a thing about Cantor.

An interesting perspective but it does go against the evidence of the red shift of distant galaxies.

Well for example (as I know someone who will criticise me for this) if you believe infinity and specifically an infinity of time is impossible then that leads logically to a timeless first cause (itself uncaused as it is beyond causality).

It would also be generally useful if we could rule out infinity so we could teach the kids something more useful/truthful than Cantor's infinite set theory.

Plus it would simplify science if we knew for sure that Actual Infinity is impossible. Certainly in cosmology, that would cut down the range of possible models of the universe.

Wittgenstein had a nice solution to stop bothering about these kinds of problem, basically do we ever observe infinite things? No we only observe finite things, and our concept of expansion stems from our observations of finite things that expand, so it is meaningless to apply a concept that applies to finite things to something that is not finite. — leo

Yes and you could add the fact we only ever observe finite things is strong inductive support for the proposition 'infinity does not exist'.

But even if you got cosmologists to stop talking about an infinite space that expands, that wouldn't change much in the grand scheme of things — leo

I see it as one battle in a war. I believe infinity is impossible in general. There are proofs but people don't buy the proofs. So I've settled for trying to show each instance of infinity leads to a contradiction. Infinite space is one of these instances.

Galaxies are moving apart from each other at greater than the speed of light. That is only possible if space/area is expanding.

The astronomers are telling that area is expanding. They liken the expansion of the universe to the expansion of a ballon with dots on it to represent the galaxies.

However when applied to the area no longer applys. — hachit

Why so? It applies to the metric - that can't be both expanding and infinite at the same times so it applies to space/'the area'?

Well you can imagine a finite part of the universe and visualize the metric expanding in that part, and imagine that the same goes on in every part of an universe that goes on forever. — leo

If it goes on forever, there is no room for any expansion; there is nowhere to expand to.

Then some might say "our mind is not able to grasp it all at once but that's only a limit of our mind", while others might say "something that cannot be conceived as a whole doesn't exist or is impossible". — leo

I'm of the 2nd believe. That head spinning feeling when we think of infinity is our minds choking on a very illogical concept I think.

There may be no expanding...just the illusion of expanding — Frank Apisa

What then is your explanation for the redshift of distant galaxies?

What we humans consider the universe may be expanding...but "what we humans consider the universe" may be but nothing within an INFINITE universe. — Frank Apisa

Good point. If our universe is expanding and our universe is contained in the larger universe, that means the larger universe must be expanding too hence it can't be infinite either.

I've spent years studying infinity and my conclusion is that the mathematical community have it wrong.

Its a belief called Finitism:

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

Used to be more popular than it is nowadays but there are still a few proponents (Max Tegmark, William Craig Lane) for it around.

You should have respect for and consider other peoples viewpoints; not dismiss them blindly.

You should realise it is a fact that not everything you were taught in school is correct... learn to be more skeptical and keep an open mind to new ideas.

One way to view it is to say that galaxies are staying still while there is some underlying space expanding, but no entity called "space" has ever been observed expanding or stretching, when we say space expands we're saying nothing more than galaxies move away from each other at a rate proportional to the distance between them, which doesn't require an expanding space to describe. — leo

But the metric is expanding. So we can equate the metric to space without having to resort to a believe in spacetime. And if the metric is expanding, the metric, IE space, cannot be infinite.

That is the standard definition of infinity and by it, some infinities are contained within others. — Isaac

The size of the universe is either infinite or not infinite. If it can grow, it is by definition not infinite. Only finite sized things can grow.

The DICTIONARY mathematical definition of infinity:

a number greater than any assignable quantity or countable number

There clearly only one such possible number - the existence of such a number by definition rules out any other infinite numbers. There is only one infinity.

The standard definition is:

a number greater than any assignable quantity or countable number

If there exists a number X+1, then X cannot be infinite... so by the standard definition of infinity, my argument holds.