Looks like the difference between Platonism and Constructivism. If you think mathematical objects are real and have existence beyond humans calculating or proving them, then infinity is actual. If you don't, then it's only potential, because we'll never add all the way up to infinity.

Formally speaking, actual infinity merely refers to the axiom-of-infinity. But this isn't a good answer, because this does not account for the controversial motives for introducing the axiom.

The underlying dilemma is the result of different interpretations of the informal sign "..." used to denote partially elicited sets, and how these different interpretations lead to different conclusions concerning the very meaning of a set, including what sort of sets are admissible in mathematics.

Ordinarily, in statements such as {0,1,2,3,...}, the sign "..." is used to state that the "set" refers to a rule (as in this case, the rule of adding one and starting from zero), as opposed to an actually completed and existent body of entities. This is synonymous with potential infinity, that appeals to one's temporal intuitions regarding a process whose state is incremented over time.

In other cases such as "my shopping list is {Chicken,wine,orange juice,...} ", the dots might denote either

i) an abbreviation for a particular, finitely describable list that is already existent, but only partially described on paper

or

ii) An indication that a list is abstract and only partially specified, that the reader is invited to actualize for himself via substituting his own items, or rule of extension.

or

iii) a mystical sign, referring to "actual infinity" in a sense that is empirically meaningless, physically useless and logically a mere piece of syntax, but which nevertheless has psychological value in causing giddy vertigo-like sensations in true-believers when they contemplate the unfathomable.

Unfortunately, because "..." is informal notation with at least three completely distinct operational uses in addition to having private psychological uses, people continue to conflate all of these uses of the dots, causing widespread bewilderment, philosophical speculation and moral panic up to the present day.

Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity. In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {. — TheMadFool

Technically, I think that it should be #{1,2, 3,...} or card({1,2, 3,...}) or |{1,2, 3,...}| for actual infinity (cardinality symbols).

1,2, 3,... is just a sequence and not a set.

sequence: Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters.

In fact, there is another notation that is very close to set and sequence: a tuple or n-tuple: (1,2, 3).

tuple: In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.

Now, to confuse the hell out of everybody, the arguments of a function are deemed a tuple, but the typical notation for variadic functions (=with variable number of arguments) is f(a,b,c, ...), while the use of the ellipsis "..." is forbidden in tuples.

Furthermore, all these things are almost the same, with just a minute subtlety here and there ... ;-)

Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here?

Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool

Yes, I think it is. It is certainly what Wikipedia says..

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential,[4] but not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist physically in nature.

Of course, as it says, any representation as to whether physical infinite exists in the real, physical world is obviously out of scope in mathematics.

The difference is presentation versus application; simulation versus stimulation.

In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {. — TheMadFool

Actual infinity is not possible; there could always be more. Infinite is not an amount or an extent completed or capped, as extant, as that can't happen.

In other words there is a largest number and infinity doesn't exist. — TheMadFool

But you will never arrive at this number, because in order to do so you'd have to take an infinite number of steps. Which is why we call it an infinity.

Don't look at infinity as a number (it's not). Look at it something like this. Infinity represents something that can get as big as you want. The whole idea of limits is based on this concept. Taking a limit of a number and assigning it a value would mean you can get as close as you want to that value (doesn't mean you can actually get that value).

to take an infinite number of steps

If l say, l have taken 100 steps, l am using the word "steps" in a different sense, to mean a numerical quantity.
When l say infinite number of steps, the word "steps" specifies the nature of steps ( they do not end) in such a system, it does not refer to a numerical quantity.
Therefore we cannot such a definition for infinity.
It is very difficult to define infinity using any concept other than infinity itself. Hence it is often circular, self referential.

In maths, how would you interpret limits that do not exist to those that exist. For example, lim x-->o ( 1/x )does not exist but lim x--> 1 ( x-1/x^2-1) . The second one has removable singularity. Somehow we can assign useful value when using infinity but not always,so there are problems sometimes

Infinity doesn't exist as a number but a concept and l think even the concept has faults.

In other words there is a largest number and infinity doesn't exist. — TheMadFool

Without the axiom of infinity, a concept of actual infinity is not viable. That is obviously also the reason why the axiom was introduced. Otherwise, there would simply be no need for it.

The use of actual infinity is not even permitted in mathematics without axiomatizing it first. Therefore, it is perfectly ok for you to reject the axiom, but then you can also not make use of any of its consequents.

Since it is the sixth axiom in ZFC, you cannot make use of ZFC either. You will need to use an alternative set theory (of which there are actually many).

One possible problem could be that you cannot make use of any of the large number of theorems that rest on ZFC, unless they do not make use the axiom of infinity. However, it is a lot of work to weed through all of that, because it requires verifying their proofs. When rejecting ZFC, a lot of things that you would do in set-theoretical context will now be incompatible with mainstream set theory. Welcome to Hassle-land where everything that would have been simple, now becomes complicated!

Thanks all for your comments.

The axiom of infinity. So an infinite set is postulated to exist.

My statement about the infinite set of natural numbers was poorly worded.

What I should have said is that a largest natural number exists by the following argument:


Let's look at the sequence of natural numbers which I think is the "simplest" infinity we can talk about.

Natural numbers: 1, 2, 3,...

Observe how successive numbers "increase"

a) 1 to 2 the quantity has doubled (2 = 2 × 1)
b) 2 to 3 : (3 = 1.5 × 2)
c) 3 to 4 : (4 = 1.33... × 3)
d) 4 to 5 : (5 = 1.25 × 4)
.
.
.as you can see the factor (numbers in bold) is decreasing and approaching a limit which is 1. Look at larger numbers below:

e) 9999 to 10000 : (10000 = 1.001... × 9999)
f) 99999 to 100000 : (100000 = 1.0001... × 99999)

The pattern suggests that eventually there will be two very very large numbers A and B such that:

1. B = A + 1 (B is the next number we getting by adding 1 to A)
2. B = A × 1 = A (the pattern I showed you suggests that 1 is the limit of the factor by which a number increases in bold)

In other words there is a largest natural number.

The set of natural numbers does not have an upper bound, so it will always have a number that is smaller than another number. In other words, there is no largest number. If you disagree with the axiom that a set can have infinite elements, then it is possible to say that there is a certain largest number in a set but otherwise no.
The problem with axiom of infinity is that it fails to fall in one of the two categories. Intension and extension.

intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used.

This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition

Some logician view that infinite extensions are meaningless as extensions must be complete in order to be well defined, so infinity cannot be defined by extensions. ( They reject Cantors proof too )
The problem with definition using intention is that they are circular.

I'm confused by the distinction actual vs potential infinity? — TheMadFool

There's a straightforward and unambiguous mathematical distinction.

The inductive axiom of the Peano axioms say that whenever n is a number, n + 1 is a number. So we have 0, and 1, and 2, and 3, ... [The fact that 0 is a number is another axiom so we can get the induction started]. However we never have a "completed" set of them. In any given application we have as many numbers as we need; but we never have all of them assembled together into a single set.

The axiom of infinity says that there is a set containing all of them.

So with the Peano axioms we may write: 0, 1, 2, 3, ...

With the axiom of infinity we may write: {0, 2, 3, ...}.

The brackets mean that there is a single completed object, the set of all natural numbers. That's Cantor's great leap. To work out the mathematical consequences of completed infinity.

I'm sure from a philosophical point of view there may be some quibbles. But this is how I think of it. the axiom of mathematical induction gives you potential infinity. The axiom of infinity gives you completed infinity.

Note that even with potential infinity, there are still infinitely many numbers. It's just that we can't corral them all into the barn. In fact in Peano arithmetic, the collection of all the natural numbers is a proper class. This is a good way to visualize what we mean when we say that a given collection is "too big" to be a set.

Axiom of infinity. That's as subtle as a gun in your face I guess. I don't know. Am I making sense here? — TheMadFool

Yes perfect sense. The axiom of infinity is a humongously ambitious claim for which there's currently no evidence in the real world. It's a bold statement. On the other hand without it, we can't get a decent theory of the real numbers off the ground. So the ultimate reason to adopt the axiom of infinity is pragmatic. It gives a much more powerful theory. Whether it's "true" in any meaningful sense is, frankly, doubtful.

There are two different domains of discussion: (1) mathematics itself and (2) philosophy of mathematics.

(1) MATHEMATICS ITSELF

(There are forms of mathematics other than classical set theoretic mathematics, but for brevity by 'mathematics' I mean ordinary classical set theoretic mathematics.)

In mathematics we don't ordinarily think in terms of a noun 'infinity' but instead of the adjective 'is infinite'. There is no object (abstract of otherwise) named by 'infinity' (setting aside in this context such things as points of infinity in the extended real system). Rather the adjective 'is infinite' holds for some sets and not for others.

Formal definitions of 'finite' and 'infinite':

A set S is finite if and only if S is in one-to-one correspondence with a natural number.

A set S is infinite if and only if S is not finite.

In mathematics itself there is not a formal set theoretic notion of 'potentially infinite'. Mathematics instead proceeds elegantly without undertaking the unnecessary complication of devising a formal definition of 'potentially infinite'.

(1) PHILOSOPHY OF MATHEMATICS

In the philosophy of mathematics, the distinction between actually infinite and potentially infinite might be described along these lines:

Actually Infinite. An actually infinite set is an object (presumably abstract) that has infinitely many members. The set of natural numbers is an actually infinite set.

Potentially Infinite. There are some philosophers or commenters on mathematics who do not accept that there are actually infinite sets. So for them there is no set whose members are all the natural numbers. Instead these commenters refer to processes that are always finite at any point in the execution of the process but that have no finite upper bound, so that for any step in the execution, there is always a next step available. For example, with counting of natural numbers, only finitely many natural numbers are counted at any given step, but there is always a next step allowed.

In constructive mathematics (not classical mathematics), perhaps, with research, one can find formal systems with a formal definition of 'potentially infinite'. But I would bet that any such system would be a lot more complicated and more difficult to work within than classical mathematics. This is the drawback of the notion of 'potentially infinite'. One can talk about it philosophically, but it takes a lot more work to devise a formal system in which 'potentially infinite' is given an exact, formal definition.

Looks like the difference between Platonism and Constructivism. — Marchesk

It is not necessary to adopt platonism to accept that there are infinite sets. One may regard infinite sets as abstract mathematically objects, while one does not claim that abstract mathematical objects exist independently of consciousness of them.

it should be #{1,2, 3,...} or card({1,2, 3,...}) or |{1,2, 3,...}| for actual infinity — alcontali

No, that is not required. (1) There are infinite sets that are not cardinals. (2) Let w (read as 'omega') be the set of natural numbers. So w = {x | x is a natural number}. That is what is meant by {0 1 2 ...} (I drop unnecessary commas). And w itself is a cardinal, and for any cardinal x, we have card(x) = x anyway.



Here is an explication of 'set', 'tuple', 'sequence', 'multiset' in (set theoretic) mathematics:

Everything is a set, including tuples, sequences, and multisets.

A tuple is an iterated ordered pair.

Definitions:

{p q} = {x | x = p or x = q}

{p} = {p p}

<p q> = {{p} {p q}}

Then also, for example, <p q r s t> = <<<p q> r> s> t>

S is a sequence if and only if S is a function whose domain is an ordinal.

S is a finite sequence if and only if the domain of S is a natural number. (There is an "isomorphism" between tuples and finite sequences. For example: The tuple <x y z> "encodes the same information" as the sequence {<0 x> <1 y> <2 z>}.)

S is a denumerable sequence if and only if the domain of S is w.

S is a multiset if and only if S is of the form <T f> where f is a function whose domain is T and every member of the range of f is a cardinal. (So f "codes" how many "occurences" there are of the members of T in the multiset.)

It is very difficult to define infinity using any concept other than infinity itself. Hence it is often circular, self referential. — Wittgenstein

There is no circularity in the set theoretic definition of 'is infinite'.

Without the axiom of infinity, a concept of actual infinity is not viable. — alcontali

Depends on what you mean by 'viable'. There is a set theoretic definition of 'is infinite' without the axiom of infinity. The axiom infinity implies that there exists a set that is infinite, but we don't need the axiom just to define 'is infinite'. I think you were pretty much saying that yourself, but I wish to add to it. Indeed, we agree that dropping the axiom of infinity makes an axiomatic treatment of mathematics extremely complicated.



Your claimed proof that there is no infinite set is not recognizable as a proper mathematical argument but instead proceeds by hand waving non sequitur.

The set of natural numbers does not have an upper bound, so it will always have a number that is smaller than another number. — Wittgenstein

No, there is no natural number smaller than the natural number 0. So maybe you meant that for any natural number n there is a natural number greater than n.

The problem with axiom of infinity is that it fails to fall in one of the two categories. Intension and extension.

"intensional definition gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used."

"This is the opposite approach to the extensional definition, which defines by listing everything that falls under that definition." — Wittgenstein

That is irrelevant because the axiom is not a definition and does not need to meet any standards of definitions. Also, we have to distinguish between two different notions of extensional/intensional. Aside from yours, there is the notion of extensionality that applies to set theory: Sets are extensional because they are determined solely by their members. That is, S = T if for all x, x is a member of S if and only if x is a member of T. And it doesn't matter whether a set is described by what you call 'intension' (such as {x | x has property P}) or, for finite sets, by finite listing in braces. For example, {x | x is a natural number less than 3} = {0 1 2}. Of course, infinite sets don't have listings such as {0 1 2}, but that does not vitiate that they exist.

Some logician view that infinite extensions are meaningless as extensions must be complete in order to be well defined, so infinity cannot be defined by extensions. ( They reject Cantors proof too )
The problem with definition using intention is that they are circular. — Wittgenstein

Maybe there are such logicians, but even constructivists accept the proof of Cantor's Theorem and Cantor's proof of the uncountability of the reals.

And there is no circularity in the definitions of set theory. Mathematical definitions are not circular (that is, if a purported definition is circular then somewhere in the formulation of the purported definition there is a violation of the formulaic rules for mathematical definition).

in Peano arithmetic, the collection of all the natural numbers is a proper class. — fishfry

I wouldn't state it that way. If we mean first order Peano arithmetic (PA), then there are not in PA definitions of 'set', 'class', and 'proper class'. Meanwhile, in set theory, the domain of the standard model of PA is a set.

This post by TheMadFool was supposed to rolled into this thread:

[start quote of post]

This is a question from an elementary math book:

u = u + 1.
(i) Find the value of u
(ii) What is the difference between nothing and zero?

If you try and solve u = u + 1 you'll get 0 = 1 (subtracting a from both sides)

0 = 1 is a contradiction. So u is nothing. u is NOT zero. u is nothing.

Why?

Take the equation below:

e + 1 = 1

Solving the equation for e gives us e = 0. The same cannot be said of u = u + 1 our first problem.

So given the above equations ( u = u + 1 AND e + 1 = 1) we have the following:

1) u is NOTHING. u is NOT zero
2) e = zero

What's the difference between NOTHING and zero?

My "explanation" is in terms of solution sets.

The solution set for u = u + 1 is the empty set { } with no members
The solution set for e + 1 = 1 is {0} with ONE member viz. zero.

There's another mathematical entity that can be used on the equation u = u + 1 and that is INFINITY.

INFINITY + 1 = INFINITY

So we have:

a) u is NOTHING
b) u is INFINITY

Therefore,

NOTHING = INFINITY

Where did I make a mistake?

Thank you.

[end quote of post]

(1) What math book is that? What is the context? What does the variable 'u' range over? What specific operation does '+' stand for?

(2) There is no mathematical object named 'infinity' (unless it's something like a point of infinity in the extended real system - and in a context like that, the operations of addition and subtraction have special modified formulations that avoid such contradictions). And if infinite sets are meant, then operations such as cardinal addition or ordinal definitions are formulated so that they may not be confused with the operations of addition on natural numbers or on real numbers.

(3) Your "nothing = infinity" is just wordplay. As mentioned, there is not an object named 'infinity'. And 'nothing' also is not the name of a mathematical object. To say something like "nothing is not equal to itself" is not saying that there is an object named 'nothing' that has the property of not being equal to itself. Rather, it means that there is no object that has the property of not being equal to itself. So then putting an equal sign between 'nothing' and 'infinity' is nonsense.

Everything is a set, including tuples, sequences, and multisets. — GrandMinnow

There is an "isomorphism" between tuples and finite sequences. For example: The tuple <x y z> "encodes the same information" as the sequence {<0 x> <1 y> <2 z>}. — GrandMinnow

What would be the operator in the isomorphism? Otherwise, without such operator, isn' it just a bijection? It is just a mapping between two sets, no?

Still, in my impression, the definition for morphism may be a bit ambiguous because in category theory they do not really seem to insist on the presence of such operator, while in abstract algebra they absolutely do.

By the way, I find abstract algebra much more accessible than certainly the deeper caves of category theory. It is only when they sufficiently overlap that it is clear to me ...

:ok: :up:

I wrote "isomorphism" in scare quotes because I don't mean an actual function. I mean that tuples and sequences are "isomorphic" in that you can recover the order from one to the other and vice versa. This can be expressed exactly, but it's a lot of notation to put into posts such as these. Anyway, the general idea is obvious and used in mathematics extensively.

Your claimed proof that there is no infinite set is not recognizable as a proper mathematical argument but instead proceeds by hand waving non sequitur — GrandMinnow

Thank you and can you be more specific. There are quite a number of steps I went through in my "proof".

The final steps in my proof:

1. b = a + 1 (just like 3 = 2 + 1)
2 b = a × 1

but as @Echarmion said I think a = b = infinity. Math breaks down at both ends of the whole number line: at zero and at infinity.

This post got out of sequence. I put the text in my next post.

(1) What math book is that? What is the context? What does the variable 'u' range over? What specific operation does '+' stand for? — GrandMinnow

A very simple text. I'm quite certain there's very little ambiguity with the concepts I used.

2) There is no mathematical object named 'infinity — GrandMinnow

Axiom of infinity?

(3) Your "nothing = infinity" is just wordplay. — GrandMinnow

How is it "wordplay"?

The solution set for a = a + 1 is the empty set { } with no members. In different words a is NOTHING, not even zero
NOTHING = NOTHING + 1

INFINITY = INFINITY + 1

Oh I see now. They may not be the same thing but just two different objects that behave in the same way. Thanks.

From what axioms, definitions, and rules of inference do you argue that? — GrandMinnow

Sorry if this puts you off but what axioms would be necessary for the existence of natural numbers and the basic mathematical operations of + and ×? I begin from these

I wrote "isomorphism" in scare quotes because I don't mean an actual function. I mean that tuples and sequences are "isomorphic" in that you can recover the order from one to the other and vice versa. This can be expressed exactly, but it's a lot of notation to put into posts such as these. Anyway, the general idea is obvious and used in mathematics extensively. — GrandMinnow

Oh, yes, agreed, it slipped my mind. It is indeed not just a set. Unlike in sets, the actual order of elements is also a piece of information that sequences and tuples carry. So, it is indeed more than a mapping between orderless sets.

A very simple text. — TheMadFool

What is the name and author of the text?

Axiom of infinity? — TheMadFool

The axiom of infinity is not a mathematical object named 'infinity'.

Moreover, the axiom of infinity itself is a finite mathematical object, as it is a finite string of symbols in a formal language.

How is it "wordplay"? — TheMadFool

I explained explicitly in my post.

Oh I see now. They may not be the same thing but just two different objects that behave in the same way. — TheMadFool

No, you just made the same mistake I pointed out the first time.

what axioms would be necessary for the existence of natural numbers and the basic mathematical operations of + and ×? I begin from these — TheMadFool

There are lots of different axiom systems for such things. For example, set theory. The existence of natural numbers is proven in set theory (even without the axiom of infinity). The existence of the set of natural numbers is proven in set theory (with the axiom of infinity). The operations of addition and multiplication are also definable and proven to exist in set theory.

Set theory proceeds from formal axioms, formal definitions, and formal rules of inference. Your argument has no apparent basis in those axioms, definitions, and rules. So I ask you what, exactly, are your axioms, definitions, and rules. Without specifying them, your argument, using such verbiage as "this pattern suggests" and then the non sequitur "in other words there is a largest natural number" is nonsensical handwaving, also known as 'waffle'.

Moreover, not just axioms, but ordinary mathematical common sense endows us with the understanding that there is no greatest natural number. Suppose there were a greatest natural number n. Then n+1 is greater than n. So n is not, after all, a greatest natural number.

This can be expressed exactly, but it's a lot of notation to put into posts such as these. — GrandMinnow

I've got a question about infinite cardinalities. The following set of sets is an element of the powerset of real numbers:

{{1.2323,343.3333},{344.2,0,34343.444,6454.6444},{2323.11,834.33},{},{5 12.1,99.343433}}

So, any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers, i.e. beth2.

Now, regular languages cannot match wellformedness. So, things like matching embedded braces { } is out of the question. But I just concocted a set notation that does not use wellformedness:

[
1.2323 343.3333
344.2 0 34343.444 6454.6444
2323.11 834.33

5 12.1 99.343433
]

It is the same information as above, but in another notation. This notation is regular and can be successfully matched by a regular expression. I tried it at the test site https://regex101.com. The regex looks like this:

Since this expression successfully matches sets of sets of real numbers, can I say that it is the membership function of a set with cardinality beth2, i.e. 2^2^beth0 ?

If that makes sense, then it would be a witness to the claim that regular expressions can describe sets of which the cardinality exceeds that of the continuum, i.e. uncountable infinity.

The following set of sets is an element of the powerset of real numbers: — alcontali

No, that set is a member of the power set of the power set of the set of real numbers.

And I don't understand the rest of your post, starting with "any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers"