## Musings On Infinity

• 2.1k

- I think potential infinity (calculus) is obviously very useful in maths and science
- I think actual infinity has no useful applications
• 1.6k
I think actual infinity has no useful applications

And yet, merely as a concept, it enhances our thinking, or at least it can enhance our thinking.
• 2.1k
Agreed, actual infinity is a usual mental aid.

The problems can up start once you try to do anything logical with it. It's not a logical concept so it leads to paradoxes. Particularly if you insist (like set theory does) that infinity is measurable and has a size then it leads to paradoxes.

It is also a problem that the inclusion of actual infinity in set theory legitimatises actual infinity and then cosmologists and other scientists come along and build actually infinite models because they think actual infinity has a sound basis in logic when it does not.
• 2.9k
The problems can up start once you try to do anything logical with it. It's not a logical concept so it leads to paradoxes. Particularly if you insist (like set theory does) that infinity is measurable and has a size then it leads to paradoxes.

On the number line between zero and one: are there more rational than real numbers? Or the other way round? To know, you have to have a way of quantifying both. How do you go about these?
• 2.1k
On the number line between zero and one: are there more rational than real numbers? Or the other way round? To know, you have to have a way of quantifying both. How do you go about these?

I would not use bijection to arrive a conclusion about a sets size; bijection claims that there are the same number of rationals as naturals so it is clearly wrong (naturals are a proper subset of the rationals so bijection is giving a wrong results).

I would recognise that infinity has no size so is not measurable.

I would recognise that what Cantor was trying to do was to assess the size of infinity (which is impossible). Instead I'd look to compare the density of rationals and reals on the real number line. Bijection is one possible way of measuring density but its frankly pretty useless: we have aleph-naught and aleph-one but there is nothing else!
• 189
This is a nonsensical definition: for instance, it claims the even numbers are the same size as the natural numbers (as there is a one-to-one correspondence between the two). But the even numbers are a proper subset of the natural numbers. If either had a size, the size of the natural numbers must be greater than the size of the even numbers.

Well, the problem is to give a definition of "size" for sets that you cannot count. And we want this definition to have some "reasonable" properties:
1. it should be the number of elements if the set is finite
2. it should be applicable to both countable and uncountable sets
3. any uncountable set should be bigger than any countable one
4. (as you required) if A is a proper subset of B, than A should be smaller than B

- To satisfy 3 you have to add an extra element (that we call "infinite") to the integers as "size" of uncountable sets, because all integers are just "taken", and by definition we require this element to be greater than every natural number.

- To satisfy 4 you have to add a lot more elements of the "infinite" kind, that have to be all different between each-other.

I don't know if it's possible to define a "size" that respects these four properties, but even if it is, you cannot do it with only one "infinite" size.

If instead you drop the property 4, you can require (by definition) that two sets have the same size if there exists a bijection between their elements. This is uglier, but not contradictory: it's only an equivalence relation between the sizes of the sets. You could even require that all the sizes between 10 and 20 are the same, and all other sizes behave normally, and that wouldn't be contradictory either.
The point is that this is only a definition: the point is not if it is "true" or not, but if it is not contradictory and if it can be used to build "interesting" theorems.
• 189
If you use a reasonable definition of infinity: ‘A number bigger than any other number’ then it is clear that there could only be one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity

True. But you can't use such number system (integers plus "infinite") to define a "size" of sets that respects the properties 1 to 4.
• 2.1k
Well, the problem is to give a definition of "size" for sets that you cannot count

I don't see the need to assign a size to something that is unmeasurable - it leads to nothing useful, just paradoxes.

I think it would have been better if it had been recognised by Cantor that there are two different sorts of set - finite and infinite. An infinite set is not a-kind-of finite set and vice-versa; as can be seen by the two object types having very different properties:

- A finite set has a completely defined list of members. An infinite set does not have this property

- An infinite set does not have a cardinality property: cardinality or size implies the ability to measure something. Infinity is by definition unmeasurable so infinite sets have no cardinality/size property

Instead Cantor just made up fictitious numbers (transfinites) for the non-existence cardinality property of infinite sets. I'm an ex-computer programmer and what Cantor did is regarded in software as a cardinal sin (pardon the pun).

A finite set can be formed in reality. Infinite sets are not part of reality and are just a conceptual thinking aid. I think maths should not try to treat the two the same.
• 189
1.Let S be the set of all sets, then |S| < |2^S|
2. But 2^S is a subset of S, because every set in 2^S is in S.
3. Therefore |S|>=|2^S|
4. A contradiction, therefore the set of all sets does not exist.

What is wrong with this ‘proof’?
The conclusion could even be that "the measure of the set of all sets does not exist" this is an assumption too.
• 2.9k
we have aleph-naught and aleph-one but there is nothing else!

Naught and c; whether c equals 1 is an open question. And nothing else? That's just plain wrong.
• 2.1k
Sorry, using 2^S to denote the power set of S. The proof I gave is meant to show that the set of all sets does not exist. I maintain that it is the cardinality of the set of all sets that does not exist.
• 2.1k
Naught and c; whether c equals 1 is an open question. And nothing else? That's just plain wrong.

Well how come nearly everything gets classified as aleph-naugh:

the set of all square numbers, the set of all cubic numbers, the set of all fourth powers, ...
the set of all perfect powers, the set of all prime powers,
the set of all even numbers, the set of all odd numbers,
the set of all prime numbers, the set of all composite numbers,
the set of all integers,
the set of all rational numbers,
the set of all constructible numbers (in the geometric sense),
the set of all algebraic numbers,
the set of all computable numbers,
the set of all definable numbers,
the set of all binary strings of finite length, and
the set of all finite subsets of any given countably infinite set.

Then there is aleph-one the continuum. No-one has ever found any other aleph number (that was not generated from a previous aleph).

What use is it a measurement when it is so crude a measurement?
• 189
↪Mephist Sorry, using 2^S to denote the power set of S. The proof I gave is meant to show that the set of all sets does not exist. I maintain that it is the cardinality of the set of all sets that does not exist.

OK, I didn't understand at first that you wanted to use it as a "fasle proof". I agree, this proof is surely not acceptable for a number or reasons. First of all, we know from Russell that we should be very careful when proving something about the set of all sets, because the "naive set theory" is contradictory. So we should use a formalization of set theory as Zermelo Fraenkel Set theory, and complete formal proofs in ZF Set theory are not so simple..
• 189
It is the assumption that infinite sets are measurable that invalidates naive set theory. ZF set theory is patchwork of hacks that tries to cover all the the holes and fails - the solution is to acknowledge infinite sets do not have a cardinality / size.

Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not inconsistent!
• 189
↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:

∞ + 1 = ∞

This assertion says in english:

’There exists something that when changed, does not change’

Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested!
• 2.9k
Well how come nearly everything gets classified as aleph-naugh:

To all of us: discussion with Devans99 is a waste of time. This careless statement of his above is just a small sliver of his nonsense. And done.
• 2.1k
Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not inconsistent!

Maybe I'm using inconsistent in the wrong way; set theory is not inconsistent within itself, but to me, Galileo's paradox and the other results bijection give are inconsistent with common sense. Especially this inconsistency with logic is noticeable for transfinite arithmetic.

Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested!

This document is reasonable (see page 23):

• 1.5k
- An infinite set does not have a cardinality property: cardinality or size implies the ability to measure something. Infinity is by definition unmeasurable so infinite sets have no cardinality/size property
But isn't an infinite set bigger than any finite set? Doesn't that imply size, even if obviously you cannot measure it like a finite set?

And what is your problem with the reductio ad absurdum proof of there not being a bijection between the natural numbers and the reals?

I would not use bijection to arrive a conclusion about a sets size; bijection claims that there are the same number of rationals as naturals so it is clearly wrong (naturals are a proper subset of the rationals so bijection is giving a wrong results).
How do you come to this conclusion? Give a proof that this isn't so. Because the proof of this being so is for me quite understandable (if I remember it correctly): you can well order the rationals, hence there is a bijection between the rationals and the natural numbers.

To all of us: discussion with Devans99 is a waste of time.
Yeah maybe, you cannot just say that set theory is wrong. It would be similar that I would simply declare quantum physics wrong in physics. You would actually need to give a proof of it in mathematics.
• 2.1k
But isn't an infinite set bigger than any finite set? Doesn't that imply size, even if obviously you cannot measure it like a finite set?ssu

An infinite set is unmeasurably bigger than a finite set. An infinite set therefore has no size.

Or: size is an integer property. Infinity is not an integer, if it were, it would be an integer X greater than all other integers. But X+1>X. So infinity is no sort of size or number.

Yeah, you cannot just say that set theory is wrong. You would actually need to give a proof of it in mathematics.ssu

I'll define infinity as ‘A number bigger than any other number’ then it is clear that there can be only one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity.

That disproves the assertion of multiple infinities in set theory. From that we can deduce that:

∞+1=∞
implies
1=0

IE infinity is not a number... disproving the transfinites from set theory.
• 1.5k
An infinite set is unmeasurably bigger than a finite set. An infinite set therefore has no size.
Ok, let's think about what you said here.

So infinite sets are unmeasurably bigger. Fine, they are bigger, we agree with this.

How then couldn't you simply say that infinite sets have an unmeasurably bigger size? Meaning that they do have a size, but the size is obviously unmeasurable, because (accurate) measuring is something that you can do in the finite realm.
• 1.5k
I'll define infinity as ‘A number bigger than any other number’ then it is clear that there can be only one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity.
So, could it be then there would be Absolute Infinity?
• 2.1k
Meaning that they do have a size, but the size is obviously unmeasurablessu

If the size is unmeasurable then it is not a size. Size has to be an integer.

So, could it be then there would be Absolute Infinity?ssu

In our minds yes but it can't exist as a consistent mathematical object:

∞+1=∞
implies
1=0

And it can't exist in the real world:

- Creating anything infinity large is impossible; not enough time / would never finish
- Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller
- Spacetime is a creation so nothing in it is infinite
• 1.5k
If the size is unmeasurable then it is not a size. Size has to be an integer.
Aha!

This is the definition that you base your thinking. An integer shows size, yes, but size doesn't have to be an integer! Just to take one definition:

In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter"

Let's remember (again) the heap of rock pebbles. A heap is more than one, but it's not exact. Yet it can perfectly perform the task of defining size. Just as you can have a < b < c , which naturally do define size, but don't give 'exact sizes' like the bijection a +1 = b. Now a < b < c isn't a bijection, yet it does define obviously size!

(And we already had the discussion about how impossible is it to count the natural numbers from 1 to googolplex. Of course, the concept of size is often applied to ideas that have no physical reality, just like in mathematics.)
• 2.1k
A heap is more than one, but it's not exact.ssu

A heap is an estimated size - looking at the heap is an act of rough measurement of size. That is possible only with finite objects; you cannot estimate the size of infinity; so I maintain infinity is unmeasurable so has no size.
• 1.5k

I just have to disagree with this.

You don't have to have the ability to make a bijection to have size, a < b does give a size comparison and size doesn't somehow evaporate without a bijection. The ability to measure accurately something exactly simply isn't a precursor for size to exist. You are creating your own math so good luck with that.
• 2.1k

"Size is the magnitude or dimensions of a thing. Size can be measured as length, width, height, diameter, perimeter, area, volume, or mass.

In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter". Size is determined by the process of comparing or measuring objects, which results in the determination of the magnitude of a quantity, such as length or mass, relative to a unit of measurement. Such a magnitude is usually expressed as a numerical value of units on a previously established spatial scale, such as meters or inches."

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

So you specifically compare two objects; one of which is a ruler, the other being the object to measure and derive a size. That is not possible with infinity (ruler not long enough).

Also note that size determination results in a 'magnitude' - infinity is not a magnitude.
• 189
↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:

∞ + 1 = ∞

This assertion says in english:

’There exists something that when changed, does not change’

I read the book that you suggested me (thanks for the link!)
I think that the "trick" that avoids inconsistency is that the inverse of infinite is not unique.

From wikipedia (https://en.wikipedia.org/wiki/Cardinal_number):
Assuming the axiom of choice and, given an infinite cardinal σ and a cardinal μ, there exists a cardinal κ such that μ + κ = σ if and only if μ ≤ σ. It will be unique (and equal to σ) if and only if μ < σ.

So, choosing μ = σ = ∞, there exists a cardinal k such that ∞ + k = ∞
k will be unique if and only if ∞ < ∞. But ∞ < ∞ is false, then k is not unique.

So, it's impossible to deduce that ∞ - ∞ = 0, and it's not allowed to use the expression ∞ - ∞ as if it was a definite cardinal number.

To derive 1 = 0 from ∞ + 1 = ∞ you should subtract ∞ from both sides of the equation, but
"1 + ∞ = ∞" does not imply "(1 + ∞) - ∞ = ∞ - ∞" because ∞ has not an unique inverse.

By the way, "∞ + 1 = ∞" does not mean ’There exists something that when changed, does not change’, but 'there exists something that doesn't change when you add 1 to it'
• 715
orry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets)

Oh yes ok, thanks for clarifying.

However, this idea is not mine: (https://www.youtube.com/watch?v=UvDeVqzcw4k) see at about min. 8:23

very non contradictory axiomatic theory based on first order logic has a finite (non-standard) model

Oh HOTT. Very interesting. I don't know much about it beyond the basics. I only watched the vid starting a little before the 8:23 mark and didn't find what you were quoting. I saw where he was showing that every mathematical object is a tree once you expand its set-theoretic nature into the sets that contain sets and so forth. Very nice little insight. Maybe I'll watch more of this. Very tragic about Voevodsky's premature demise. If he said there are finite models I'm sure there are!
• 2.1k

So, it's impossible to deduce that ∞ - ∞ = 0, and it's not allowed to use the expression ∞ - ∞ as if it was a definite cardinal number.

To derive 1 = 0 from ∞ + 1 = ∞ you should subtract ∞ from both sides of the equation, but
"1 + ∞ = ∞" does not imply "(1 + ∞) - ∞ = ∞ - ∞" because ∞ has not an unique inverse.

I see, thanks. If you believe there is only one infinity (like I do), then ∞ - ∞ = 0 is fairplay, leading to 1=0.
• 189
If he said there are finite models I'm sure there are!

First of all, I used the term "finitary model" meaning "a model that is built from a finitary theory", but probably it's a misused term. So, let's speak only about "finitary first order theories" with this definition: "A first-order theory is called finitary when it's expressions contain only finite disjunctions or conjunctions".

Of course I could be wrong, and I could have misinterpreted what Voevodsky says in the video, but the fact that first order set theories have a model whose elements can be put in a one-to-one relation with natural numbers is a theorem of model theory, basically due to the fact that you can always take as a model the strings that correspond to well-formed terms of the language: in other words, the theory can be interpreted as speaking about "strings of characters".

As usual, a citation from the old good wikipedia can come to the rescue :-)
"Set theory (which is expressed in a countable language), if it is consistent, has a countable model; this is known as Skolem's paradox, since there are sentences in set theory which postulate the existence of uncountable sets and yet these sentences are true in our countable model."
(https://en.wikipedia.org/wiki/Model_theory)

So, you could even say that "every mathematical object is a string of characters", or (as Godel did in his famous incompleteness theorem) that "every mathematical object is a natural number" (because he proved that every theorem can be translated in another theorem speaking about natural numbers).

This of course doesn't mean that uncountable sets "do not exist", but only that you cannot use a finitary
first order logic theory to prove that they exist.

But, at the end, you find that every formal logic based on rules and axioms has the same problem (I know, I should explain why, but it would be too long to start discussing this here). For this reason, I believe that the existence of non numerable infinite sets should be treated as a problem to be decided by physics, and not by mathematics.
I know the obvious objection: how do you make a physical experiment to check if something (like for example space-time) is made of a countable or uncountable set of points? Well, I don't know :-)
But the fact that calculus becomes much simpler and "beautiful" if it is interpreted using infinitesimals instead of limits (look for example at smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis) for me is a good indication that, in some sense, infinitesimal "objects" exist in nature, because laws of physics are written with differential equations.
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