Cantor did claim actual infinity exists: — Devans99

Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity.

How exactly can the set of naturals be potentially infinite? — Devans99

We can prescribe how you would logically go about constructing the set of all natural numbers, but we cannot actually carry that process out to its completion.

It is defined as an actual infinity (all sets are actual). — Devans99

Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality.

Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual? — aletheist

Yes I do. That's why I can't so easily sign on to the proposition that math isn't true. Some parts of math are obviously true, or actual. It's easy enough to say that abstract math isn't true in the sense of physically true. But there are truths that aren't physical. "5 is prime" is one of them. The axiom of infinity is NOT one of them. The axiom of infinity is taken as an axiom in set theory but is not (as far as we know) true or even meaningful in the actual world. But "5 is prime" IS true in the actual world.

Is that a distinction you find meaningful? Or do you not make a distinction between "5 is prime" and the axiom of infinity?

Sure, but in the quoted text, he did not claim that there is an actual set containing all the natural numbers. And his (incorrect, in our view) belief that there is an "actually infinite number of created individuals" does not somehow falsify all of his mathematical ideas about infinity. — aletheist

He believed actual infinity is possible logically and in reality. And many people are still under that impression. What we are all taught at school - the Dedekind-Cantor continuum - a line is an actual infinite set of points - is an actual infinity.

Please provide an authoritative reference for the claim that "all sets are actual." Remember, mathematical existence does not entail metaphysical actuality. — aletheist

"In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right" - https://en.wikipedia.org/wiki/Set_(mathematics)

Something that has potential but not actual existence is not well defined.

Some parts of math are obviously true, or actual. — fishfry

In my view, nothing within mathematics is actual--again, it is the science that reasons necessarily about strictly hypothetical states of affairs--and truth has to do with what is real, not just what is actual. Specifically, a proposition is true if and only if its subjects denote real objects and its predicate signifies a real relation among those objects. The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it, while the actual is that which acts on and reacts with other things.

But there are truths that aren't physical. "5 is prime" is one of them. — fishfry

We agree on this--the number 5 and the character of being prime are real, even though they are not actual.

But "5 is prime" IS true in the actual world. — fishfry

How so? Given my definitions above, perhaps what you mean is that "5 is prime" is true in the real world; in which case, again, we agree on this.

What we are all taught at school - the Dedekind-Cantor continuum - a line is an actual infinite set of points - is an actual infinity. — Devans99

Okay, but you and I agree that this now-standard mathematical definition of a continuum is philosophically faulty; in my case, because I hold that a line is not composed of an actually infinite set of points. Instead, the continuous whole is ontologically prior to any discrete parts, which only become actual when we arbitrarily mark off a finite quantity of them for some purpose.

Something that has potential but not actual existence is not well defined. — Devans99

Please provide an authoritative source for this claim, or just acknowledge that you made it up. Here is what "well-defined" means in this specific context.

• "A set is well-defined if there is no ambiguity as to whether or not an object belongs to it, i.e., a set is defined so that we can always tell what is and what is not a member of the set." (source)
• "In mathematics, a well-defined set clearly indicates what is a member of the set and what is not." (source)

By virtue of the procedure by which we could logically go about constructing the set of all natural numbers, it clearly qualifies as well-defined in the relevant sense. Even though there is a potential infinity of its members, such that we could never actually assemble the complete set, we can always easily determine whether any proposed candidate is or is not one of those members. 5 is, but 5.1 is not. 750,943,179,981,061 is, but a banana is not.

So, can you provide an example of something whose membership in the set of all natural numbers is ambiguous? That would be the only way to demonstrate that it is not well-defined.

The axiom of infinity:

$\exists \mathbf {I} \,(\emptyset \in \mathbf {I} \,\land \,\forall x\in \mathbf {I} \,(\,(x\cup \{x\})\in \mathbf {I} )).$

I believe (I'm not a mathematician so forgive me if I'm wrong) this reads ‘there exists a set I for which the null set is a member of I and for all x belonging to I, x union the set formed by x also belongs to I’.

I do not believe this is a declaration of potential infinity - it is a declaration of actual infinity. It says there exists such a set - it does not reference limits or sequences or any of the mechanisms of potential infinity. It does not say such a set exists potentially - it says to me it actually exists.

I take your point that one could perhaps interpret it as potential infinity but I do not believe this is the common / mainstream interpretation.

It says there exists such a set — Devans99

Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80.

In my view, nothing within mathematics is actual — aletheist

I admit to not understanding the distinction between real and actual as you tried to explain it in this post. But consider Internet security. Are cryptocurrencies and online security actual? They have actual (in the everyday sense) impact in the real world of our lives. But online security is based on public key cryptography, which is 100% based on abstract number theory; namely, the theory of factoring large composite numbers.

It's interested that for over 2000 years, number theory was considered beautiful but useless mathematics. It wasn't till the mid 1980's that public key cryptography was invented and then used as the basis of all online encryption and security.

This is a striking example of abstract, meaningless math becoming suddenly actual.

What say you? Again I confess I don't know what you mean by real versus actual. Can you give an example?

Again I confess I don't know what you mean by real versus actual. Can you give an example? — fishfry

I am largely employing the terminology and philosophy of Charles Sanders Peirce in all this, so I will offer a couple of his examples. If I were to hold a stone in my hand and then release it, then it would fall to the floor. This subjunctive conditional proposition represents a real law--one that is true regardless of whether I ever actually carry out the experiment that it describes. Similarly, any diamond really possesses the character of hardness, regardless of whether anyone ever actually scratches it with corundum to demonstrate that it does.

I do not know enough about public key cryptography to hazard a guess at how the distinction between reality and actuality applies to it. In general, I believe that the practical effectiveness of mathematics stems from its hypothetical nature; the key is that the idealized model must adequately capture the significant aspects of the actual situation. I am a structural engineer, so I routinely use a computer to simulate the effects of gravity, wind, earthquake, etc. on a building in order to design it such that it can be expected to remain standing once actually constructed.

When it is changed, it is not changed — Devans99

Reminds me of Hilbert's Hotel. It helps to make a distinction here - that between quantitative and qualitative change.

Since your argument is just a reworked Hilbert's Hotel Paradox I'll go with the Hotel analogy.

Imagine a hotel with infinite rooms and with an infinite number of guests a1, a2, a3,... Now imagine a new guest b1 who wants to stay in that hotel. The manager simply moves a1 to a2, a2 to a3, aN to aN+1,... and b1 gets a1's room. Notice that though the quantity, infinity is still infinity, hasn't changed, the quality has: b1 is the new guest.


Similarly for an infinite number of new guests c1, c2, c3,..., we do the following:a1 moves to a2, a2 moves to a4, a3 moves to a6, aN moves to a2N,...which frees up the odd numbered rooms for the infinite guests c1, c2, c3,... As you'll notice though there's no change quantitatively there is a qualitative change, the guests c1, c2, c3,...are new.


Since infinity is an exclusively quantitative concept, it fails to register qualitative changes but that doesn't mean no change has occurred.

To illustrate take 3 people, Tom, Dick and Harry. Now imagine Tom gets replace by Jane. We still have 3 people viz. Jane, Dick and Harry. There's no change in quantity but there's a qualitative difference viz. Jane is now in place of Tom.

Imagine a hotel with infinite rooms and with an infinite number of guests a1, a2, a3,... Now imagine a new guest b1 who wants to stay in that hotel. The manager simply moves a1 to a2, a2 to a3, aN to aN+1,... and b1 gets a1's room. Notice that though the quantity, infinity is still infinity, hasn't changed, the quality has: b1 is the new guest. — TheMadFool

My argument uses sequences of identical bananas, so that the 'quantity' and 'quality' of bananas both are constant whilst bananas are added and removed from the sequences - resulting in absurdity.

Yet again: mathematical existence does not entail metaphysical actuality. Within mathematics, the number 10^100 (1 googol) indubitably exists and is a member of the set of natural numbers; but according to physics, the total quantity of actual particles in the entire universe is only about 10^80. — aletheist

A googol-sized set could have logical existence if our universe was bigger. A greater than any number sized set does not even have logical existence (leads to absurdities so it cannot be logically sound).

A googol-sized set could have logical existence if our universe was bigger. — Devans99

By that reasoning, an infinite set could have logical existence if our universe was infinite. But logical possibility is not at all dependent on actuality, or even metaphysical possibility. So infinite sets do exist, in the strictly mathematical sense of existence.

A greater than any number sized set does not even have logical existence (leads to absurdities so it cannot be logically sound). — Devans99

It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets.

It only leads to absurdities if one insists on attempting to apply the same rules to infinite sets as to finite sets. Mathematicians have long recognized this, which is why there are different rules for infinite sets. — aletheist

The same rules apply for finite and infinite sets. And if you don't like the absurdity in the OP, see this famous example for proof that actual infinity is logically absurd:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox

Why not avoid all the redundancy by putting b1 in the next room aN+1?

The same rules apply for finite and infinite sets. — Devans99

As long as you continue to insist on this, there is nothing more for us to discuss.

When it is changed, it is not changed

Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?

So I think we have to conclude that actual infinity is not part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).

Interesting questions. I'm thinking the simple answer is that infinity is indeed part of our abstract reality. And that reality has contradiction and unresolved paradox.

Examples of abstract reality include the paradox of time, mathematics (Pi), cosmology (infinite universe theories), metaphysical phenomena, consciousness, so on and so forth..

Interesting discussion!

Examples of abstract reality include the paradox of time, mathematics (Pi), cosmology (infinite universe theories), metaphysical phenomena, consciousness, so on and so forth.. — 3017amen

I think I see what you mean, maybe you can expand? I see that Pi cannot, in our reality, ever be actualised as it has infinite digits. A perfect circle cannot exist in our reality. But a perfect circle exists as an idea in the mind (along with talking trees). Is therefore a perfect circle an illogical/impossible idea or would you call it abstract reality?

I think the example of the axiom of choice is an interesting point. It claims it is possible to select one ball from an infinite number of bins:

1. Clearly in concrete reality, it is not possible to complete a never ending task
2. It also seems illogical for the mind to allow that we can complete a never ending task
3. Yet we can imagine it so and imagine the consequences. This is maybe the abstract reality you refer to?

There is of course much paradox/contradiction associated with language/self-reference, as you probably already know. Gödel's incompleteness/infinite theory speaks to that through both language and mathematics (liar's paradox and variations of same).

Also, that's a great question viz illogical or abstract. I wonder if it is simply an " illogical abstract " that actually exists in reality. Much like the metaphysical phenomenology of how the subconscious and conscious mind work together in an illogical manner (violating the laws of bivalence/LEM).

In a similar way, one answer to that question of abstract reality, I think, is the fascination with the conundrum of being and becoming. At the risk of redundancy from another thread, the paradox of time and your notion of infinity is very intriguing:

Interesting video. I have sometimes wondered about the length of 'now' - it seems it cannot be zero length else 'now' would be nothing (nothing length zero has existence). This type of thinking leads to consideration that time maybe discrete and eternal, over which the is much controversy.

I wonder if it is simply an " illogical abstract " that actually exists in reality. Much like the metaphysical phenomenology of how the subconscious and conscious mind work together in an illogical manner (violating the laws of bivalence/LEM). — 3017amen

I think reality is strictly logical; I can find no instances of paradoxes/contradictions in reality. I think that the paradoxes/contradictions result from our models of reality rather than reality itself. Fundamentally our minds are capable of illogical reasoning but reality seems constrained to be logical only.

Our minds are part of a logical reality, but yet they are not constrained to purely logical concepts. When we think top-down about concepts, illogical concepts can surface in our mind. When we think bottom-up about concepts, these 'illusions' are often banished.

I think that 'abstract reality' might be a product of top-down thinking. It is easy to imagine that it is possible to square a circle when you think about it top-down. Yet it took 1000s of years of maths to prove it is logically impossible. So when we think about actual infinity, we are thinking about something in a top-down manner (sure something can go on forever). It is only when the concept of actual infinity is probed in detail that we see the problems - the paradoxes/contradictions show us that in fact we are imagining something illogical.

Beyond the boundary is nothingness IMO. Nothing cannot be actually infinite because it is nothing. If it is other universes then they cannot be actually infinite because it would lead to the absurdities referenced in the OP. Or see here for another example of the absurdity of actual infinity:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox — Devans99

You physically can't keep putting 10 balls in a vase, while only removing one. It's an unrealistic thought experiment.

I'm not talking about any type of container holding an infinite number of universes. It doesn't make sense to talk about some container with an infinite number of things. Maybe that is your problem.

If causation (space-time) isn't infinite, then you have to come up with an explanation as to how something came from nothing. I think that is a much more difficult problem to deal with than conceiving of infinity.

Double post

You physically can't keep putting 10 balls in a vase, while only removing one. It's an unrealistic thought experiment. — Harry Hindu

Why is it unrealistic?

If causation (space-time) isn't infinite, then you have to come up with an explanation as to how something came from nothing. — Harry Hindu

I don't believe everything came from nothing, I believe that something has permanent, atemporal existence and that something caused everything else. See here:

https://thephilosophyforum.com/discussion/7391/everything-in-time-has-a-cause/p1

I have sometimes wondered about the length of 'now' - it seems it cannot be zero length else 'now' would be nothing (nothing length zero has existence). This type of thinking leads to consideration that time maybe discrete and eternal, over which the is much controversy. — Devans99

On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value.

On the contrary--if time were discrete, then it would necessarily consist of durationless instants at some fixed interval. The fact that "now" cannot have zero duration requires time to be continuous, such that "now" has a duration that is infinitesimal--not zero, yet less than any assignable or measurable value. — aletheist

Discrete time would consist of discrete, non-zero, non-infinitesimal time slices - so they would have a duration.

On the other hand, if time was composed of infinitesimal time slices, then each fixed period of time would be composed of 1/∞=0 length time segments giving a zero total length for all elapsed intervals. Which cannot be right.

Discrete time would consist of discrete, non-zero, non-infinitesimal time slices - so they would have a duration. — Devans99

In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer.

On the other hand, if time was composed of infinitesimal time slices, then each fixed period of time would be composed of 1/∞=0 length time segments giving a zero total length for all elapsed intervals. Which cannot be right. — Devans99

The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals.

In other words, continuous lapses of time with finite duration, arranged such that each one starts when the previous one ends. Calling this "discrete time" is a misnomer. — aletheist

They are distinct, like a movie plays at 60 frames a second, each frame a time slice. There is nothing continuous about that.

The last sentence is correct, because the first sentence demonstrates a complete misunderstanding of infinitesimals. — aletheist

OK, what is the infinite sum of every possible infinitesimal:

https://www.wolframalpha.com/input/?i=limit+%28n%2Finfinity%29+as+n-%3Einfinity

It's zero. So infinitesimals cannot be the constituents of time because all time intervals would have zero length and time would not flow from one moment to the next.

They are distinct, like a movie plays at 60 frames a second, each frame a time slice. There is nothing continuous about that. — Devans99

That is exactly what I described as discrete time, not what you described. Each individual frame of such a movie corresponds to an instantaneous state, with zero duration, arranged at a fixed and finite interval of 1/60th of a second. There is no flow of time between the frames, just a leap from one to the next.

So infinitesimals cannot be the constituents of time because all time intervals would have zero length and time would not flow from one moment to the next. — Devans99

Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. The present moment includes an infinitesimal portion of the past and an infinitesimal portion of the future, which is why time does flow from one moment to the next.

If you truly want to understand the mathematics of infinitesimals, I recommend learning about synthetic differential geometry, also called smooth infinitesimal analysis. These links are to excellent brief introductions by Sergio Fabi and John Bell, respectively.

Again, this reflects a complete misunderstanding of infinitesimals. A moment of time has a duration that is not zero, but is less than any assignable or measurable value relative to any arbitrarily chosen unit. — aletheist

Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. As demonstrated in the op, ∞ leads to logical absurdities, so logically 1/∞ must be absurd too.

I see that Pi cannot, in our reality... — Devans99

With every circle,and with every other thing which has anything to do with π. With every measuring device with π marked on it (not too many of those). You're not distinguishing between number and numeral, which is typical since that distinction was made explicit to you by one the mathematicians writing here.

Infinitesimals are deeply illogical/impossible concepts and are shunned by most of maths. As demonstrated in the op, ∞ leads to logical absurdities, so logically 1/∞ must be absurd too. — Devans99

There is nothing inherently contradictory about the mathematical concept of an infinitesimal, which is not necessarily defined as 1/∞. Again, if you truly want to understand, please read one or both of the short articles that I linked. If you prefer to remain ignorant, carry on.