Musings On Infinity

Devans99

Grandi's series

I think this series tells us that infinity does not have a size and therefore infinite sets do not have a cardinality. The series is:

{ 1, -1, 1, -1, 1, -1, … }

To evaluate the infinite series requires knowledge of whether the above infinite set has an odd or even cardinality. That mathematics cannot produce (a sensible) evaluation of this series is tacit admission by mathematics that infinite sets do not have a cardinality.

What does Divide By Zero Signify?

Division by an infinitesimal is associated with infinity:

lim 1/n = ∞
n->0

Dividing by zero is however undefined. It is nonsensical to divide an object zero ways:

1/0 = UNDEFINED

Also, division being the inverse of multiplication, there is no number x such than x * 0 = 1.

What is going on here? A finitist might say the above two equations correspond to potential and actual infinity respectively:

- The limit represents potential infinity; we can potentially continue to arrive at larger and larger numbers (by setting n smaller and smaller)
- But we can never ‘complete’ the process of arriving at larger and larger numbers - we can never reach ‘Actual Infinity’ - if we try to, the expression becomes UNDEFINED - exactly what we’d expect if actual infinity does not exist.

So actual infinity is the same as 1/0 is the same as UNDEFINED.

Is There an Actual Infinity of Points on a Line Segment?

An illustration that UNDEFINED corresponds to actual infinity: it is conventional in mathematics to regard the number of points on a line segment as actually infinite. A finitist would disagree with this definition:

- Consider a line segment length 1
- The points on it have length zero (They are just logical labels on a line)
- So the length of the interval 1 divided by the length of a point 0 equals UNDEFINED number of points on the segment

The same argument applies when considering the number of real numbers in the interval 0…1, numbers have length zero, so the number of real numbers in the interval is UNDEFINED.

Devans99

Proof infinity is not a number

First, we realise if two different types of infinity existed, they would have to be larger than each other. Thats a logical contradiction, so we must of made a wrong assumption; only one kind of infinity can exist.

Then:

∞ + 1 = ∞

Leads to:

1 = 0

Which is another logical contradiction; infinity is not a number.

whollyrolling

↪Devans99

Who said infinity was a number, other than four year olds?

Devans99

↪whollyrolling

Cantor. The transfinite numbers in set theory.

whollyrolling

↪Devans99

I can see why his mathematics are "of great philosophical interest". They mean nothing and lead nowhere.

Devans99

↪whollyrolling

Agreed. He was mad:

“I have never assumed a ‘Genus Supremum’ of the actual infinite. Quite on the contrary I have rigorously proved that there can be no such ‘Genus Supre- mum’ of the actual infinite. What lies beyond all that is finite and transfinite is not a ‘Genus’; it is the unique, completely individual unity, in which every- thing is, which contains everything, the ‘Absolute’, unfathomable for human intelligence, thus not subject to mathematics, unmeasurable, the ‘ens simplicis- simum,’ the ‘Actus purissimus,’ which is by many called ‘God.’ - Cantor.

He though God was talking to him and giving him these ideas about infinity. It's all marsh gas.

Devans99

What he's says there about absolute infinity not being subject to maths... if only he'd realised that was the case for infinity full stop (there is only one infinity).

tim wood

↪Devans99

Question: Why are you so determinedly ignorant on these topics? You're playing with words that it's clear you do not understand.

Devans99

↪tim wood

Specific counter arguments rather than waffle please.

tim wood

↪Devans99

What waffle? You spout ignorance. What's the waffle there? What's of interest to me is why you do. You don't seem either ignorant not stupid, so why?

Devans99

You spout ignorance. — tim wood

Point out my instances of ignorance then...

tim wood

↪Devans99

Sorry. I and many others have played before. Not interested now.

If your only point were that infinity is not a number in the sense that 2 or the square root of 2 are numbers, you would be, so far as I know, correct. Is that all?

Mephist

First, we realise if two different types of infinity existed, they would have to be larger than each other. Thats a logical contradiction, so we must of made a wrong assumption; only one kind of infinity can exist. — Devans99

Cantor's "diagonal" theorem on the existence of an infinite hierarchy of infinities can be expressed in a quite convincing way: "for every set A, the set of all subsets of A is bigger than A".
Of course, we have to give a concrete definition of what "bigger" means:
"the set B is bigger than the set A if there isn't any function that for every element of A gives an element of B and covers all B" ( i.e. each element of B corresponds to some element of A )

The demonstration is quite easy (https://en.wikipedia.org/wiki/Cantor%27s_theorem). But there is a problem with the statement of the theorem: Russel's paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox). The concept of "set of all subsets" is contradictory.

So, Russel's idea is, basically, to avoid talking about sets, and talk only about intuitively well-defined things: functions defined on "recursive types". In this way, Cantor's theorem becomes: "for every recursive type A, the type A->Bool is bigger than A".
Put in this way, Cantor's theorem only means that it's impossible to build a "surjective" function that associates to every natural number ( "Nat" ) a second function from natural numbers to booleans ( "Nat -> Bool" ). So ( "Nat -> Bool" is bigger than "Nat" ). At the same way, "Nat -> (Nat -> Bool)" is bigger than "Nat -> Bool", "Nat -> (Nat -> (Nat -> Bool))" is bigger than "Nat -> (Nat -> Bool)", etc..

This is what it means "an infinite hierarchy of of infinite sets each one bigger than the other".
Everything is built using functions on natural numbers and very basic logic deductions, and that's the whole point: it makes sense to speak about infinite objects because they are only built by finitely defined functions. So, in a sense, from the point of view of logic, all infinites are only "potential"

fishfry

The demonstration is quite easy (https://en.wikipedia.org/wiki/Cantor%27s_theorem). But there is a problem with the statement of the theorem: Russel's paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox). The concept of "set of all subsets" is contradictory. — Mephist

You have a detailed argument for that false claim? I'd like to understand your thinking here. The set of all subsets of a given set is in no way refuted by Russell's paradox.

So, in a sense, from the point of view of logic, all infinites are only "potential" — Mephist

Math goes beyond logic. Even Russell accepted the axiom of infinity, which posits an actual infinite set whose elements include all the natural numbers. If you deny the axiom of infinity you have Peano arithmetic, which is fine as far as it goes, but does not allow a satisfactory theory of the real numbers. So you have to throw out modern physics along with most of modern math.

The solution to Russell's paradox is the axiom schema of specification. And regardless, Russell's paradox does not contradict or invalidate the powerset axiom.

Mephist

↪fishfry

From (https://en.wikipedia.org/wiki/Russell%27s_paradox):
"In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory."
Russel's paradox is about "the set of all sets that don't contain theirself", but the problem was how to limit the language (axioms and rules of logic) to ensure that no sentences of the type "the set of all sets that don't contain theirselves" are allowed. One of the solutions is type theory.
In the standard contemporary mathematics based on set theory you can't speak of the "set of all sets" because it's not a set itself, but rather you have to speak about the "class" of all sets.
If you use first order logics on the domain of real numbers, the set of all subsets of real numbers is the same thing as "the set of all sets"

fishfry

If you use first order logics on the domain of real numbers, the set of all subsets of real numbers is the same thing as "the set of all sets" — Mephist

No not at all. For example let us consider the set of all real numbers that are not members of themselves. That's a legal set formation according to the axiom schema of specification. That is, we start with a known set, the reals, and then reduce it by a predicate.

So, what is the set of all real numbers that are not members of themselves? Well, 14 is not a member of 14. Pi is not a member of pi. The cosine of 47 is not a member of the cosine of 47. In short, the set of all real numbers that are not members of themselves is ... drum roll ... the real numbers.

That's exactly how specification avoids Russell's paradox.

You have claimed that Russell's paradox invalidates the powerset axiom but I still don't follow your logic. In fact if the powerset axiom were false, I would have heard about it.

You will note that the formal expression of the powerset axiom is in fact first order. But perhaps there are some subtleties that you can elucidate in this regard.

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

ps -- The Wiki article says:

The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

Perhaps you are making a constructivist argument. It's true that the constructive powerset of the natural numbers is countable and Cantor's theorem doesn't apply. If you are making that type of argument it would be helpful to say so up front so as to not confuse the issue. If you are a constructivist and therefore don't believe in the full powerset as given by ZF, your remarks would make more sense.

pps -- Regarding my claim that the powerset axiom is first order, I refer you to the SEP article on set theory.

https://plato.stanford.edu/entries/set-theory/#AxiSetThe

ZFC is an axiom system formulated in first-order logic ...

The article then goes on to list the standard axioms, one of which of course is powerset.

Mephist

↪fishfry

OK, maybe I wanted to make it too simple :smile:
ZF Set theory is another way to limit the use of "naive set theory", in my opinion much more complicated than Type theory.
The point is that both of them are "axiomatic" theories: ZF set theory speaks about something called "set" giving a complicated set of rules on how to operate with them. But with the same set of rules the world "set" can be interpreted as a different finitely defined structure (for example as as recursively branching trees). Every non contradictory axiomatic theory based on first order logic has a finite (non-standard) model (https://en.wikipedia.org/wiki/Compactness_theorem)

Mephist

You have claimed that Russell's paradox invalidates the powerset axiom but I still don't follow your logic. In fact if the powerset axiom were false, I would have heard about it. — fishfry

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica

fishfry

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist

Of course. Perfectly correct. Perhaps you're thinking of the kind of set theory used by Frege, not by Russell. But that's a historical point and I'm not familiar enough with the specifics of Russell.

Regarding type theory, sure. No problem there. Type theory's even making a modern comeback. But I don't think I wrote a word in opposition to your remarks on type theory. You're defending a point I didn't even disagree with.

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

Oh my, no. Not at all. You should read the link you posted. There's no nonstandard finite model of ZF. Please reread the Wiki page you linked. It says nothing in support of the false claim you made. There is no nonstandard finite model of ZF. That's not what the compactness theorem says.

Mephist

Perhaps you're thinking of the kind of set theory used by Frege — fishfry

Yes, exactly. not in "Principia Mathematica" ( my mistake ).

OK, I am glad to hear that you agree :smile:

I'll add even something else, that probably everybody will cause a lot of dissent.. :smile:
In my opinion the existence of infinite sets (such as the uncountably infinite number of points in a line) is not "demonstrable" by logic or mathematics, but is more related to physics. Then, I think that axioms like the axiom of choice or the continuum hypothesis, that are logically independent from the rest of "purely combinatorial" axioms should be treated in a similar way as the Euclid's parallel postulate in euclidean geometry: they are not decidable on the base of the sole logic.

Mephist

Oh my, no. Not at all. You should read the link you posted. There's no nonstandard finite model of ZF — fishfry

Sorry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets)

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

Devans99

"the set B is bigger than the set A if there isn't any function that for every element of A gives an element of B and covers all B" ( i.e. each element of B corresponds to some element of A ) — Mephist

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.

This is what it means "an infinite hierarchy of of infinite sets each one bigger than the other". — Mephist

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.

In the standard contemporary mathematics based on set theory you can't speak of the "set of all sets" because it's not a set itself — Mephist

The set of all sets does not exist! Standard ‘Proof’:

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’?

- It is that the cardinality of the set of sets does not exist (infinite sets do not have a cardinality - infinity - is unmeasurable)
- This proof does not prove that the set of sets does not exist. This ‘proof’ is a sham
- Once it is acknowledged that the cardinality of an infinite set does not exist the whole of infinite set theory collapses like a pack of cards.

What I wanted to say is that Russel's paradox invalidates the use of "naive set theory", that is the kind of set theory used on Principia Mathematica — Mephist

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.

↪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’

Thats a straight contradiction.

tim wood

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. — Devans99

Clearly you know how many even numbers there are and you know how many natural numbers there are. And since you argue the set of natural number is the larger, then in consequence it must be possible to count the even numbers. Please do so and tell us how many even numbers there are.

Devans99

Clearly you know how many even numbers there are and you know how many natural numbers there are. And since you argue the set of natural number is the larger, then in consequence it must be possible to count the even numbers. Please do so and tell us how many even numbers there are. — tim wood

If you read any of the above, you would have gathered that I maintain infinity does not have a size so it is impossible to measure the size of infinite sets.

All we can do is induction - for any reasonably sized interval, there are more natural numbers than even numbers.

tim wood

If you read any of the above, you would have gathered that I maintain infinity does not have a size so it is impossible to measure the size of infinite sets. — Devans99

Oh but it is. They have relative sizes. Some are the same size as others, some are larger, some smaller. As it happens, it is a surprise and non-intuitive that some sets that seem as if they ought to be different sizes, are the same size.

But let's get to it. You dispute what the world accepts. The world accepts what you dispute because of the quality of the proofs offered. Now you present yours. No "maintaining," no ignorant claims. Just do it. And if you cannot, then consider taking time out to learn something about your favorite subject.

Devans99

↪tim wood

OK whats wrong with this proof?

Infinity is not a number:

if infinity was a number, it would be a number X greater than all other numbers. But X+1>X

tim wood

First of all, you have to settle down with a more careful definition of infinity. Mostly you've got it, I think, but then you misuse it, which means you haven't got it. And then you have to be careful, which you neither are nor appreciate the need to be, with the concepts applied to transfinite sets. That is, basic arithmetic functions don't work quite the same way. And so on.

So, infinity as you use it is a number, and that number plus one is still represented by that number. Now go learn why. And I think you could get a good way along by simply reading the many replies to your many posts here and elsewhere on these topics of yours, replies made in good faith to educate you, which you pay little or no attention to.

Pattern-chaser

infinity is not a number — Devans99

This isn't news to mathematicians. When the concept of infinity was invented, there was a (perceived) need for it to be integrated into mathematics. (The alternative was to leave infinity standing alone and lonely, and this (apparently) was unacceptable.) The mess you observe is the result of that 'integration'.

Devans99

And then you have to be careful, which you neither are nor appreciate the need to be, with the concepts applied to transfinite sets. That is, basic arithmetic functions don't work quite the same way. And so on. — tim wood

Basic arithmetic does not work at all with the transfinites:

∞+1=∞ implies there exists something that when changed, does not change
∞/2=∞ falls foul of the axiom 'the whole is greater than the parts'

(both cases flaunt logic)

Devans99

This isn't news to mathematicians. When the concept of infinity was invented, there was a (perceived) need for it to be integrated into mathematics. (The alternative was to leave infinity standing alone and lonely, and this (apparently) was unacceptable.) The mess you observe is the result of that 'integration'. — Pattern-chaser

That's a realistic attitude, but my personal experience is it is not shared by many mathematicians. They tend to get very defensive whenever infinity is questioned.

1. Creating anything infinity large is impossible; would never finish
2. Creating anything infinity small is impossible; no matter how small it is made, it could still be smaller
3. Space cannot be infinite because it is expanding
4. Time cannot be infinite because an infinite temporal regress is impossible
5. Only in our minds can things continue ‘forever’; in reality this would be akin to magic

If we can establish that actual infinity is not a number and is not part of our universe, there will be an opportunity to return to set theory and clean up the mess.

Pattern-chaser

↪Devans99

But infinity is a useful and meaningful concept. No matter how poorly it fits with some other stuff we invented. That's life, I suppose. :wink:

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