Actual infinities.

[Jehanne]

Cantor's proof is mathematical; of course, no one is claiming that you can enumerate all numbers of the real number line between any two finite points, only that such an infinite set exists.  The point is that WLC's claims against "actual infinities" are nonsensical.

Again, I ask what is it that is infinite?  I think that as soon as you define the what, you lose the infinity.  It is largely a trick of non-definition.

Professor Kenneth Rosen, in his very popular textbook series on Discrete Mathematics, defines an "infinite set is one that is not finite".  Or, in other words,

When we've been there ten thousand years,
Bright shining as the sun,
We've no less days to sing God's praise,
Than when we first begun.

An "actual infinite", no?

P.S. Here's Wikipedia's article on Cantor's proof:

https://en.wikipedia.org/wiki/Cantor%27s...l_argument

[RoadRunner79]

Again, I ask what is it that is infinite?  I think that as soon as you define the what, you lose the infinity.  It is largely a trick of non-definition.

Professor Kenneth Rosen, in his very popular textbook series on Discrete Mathematics, defines an "infinite set is one that is not finite".  Or, in other words,

And the question still remains, a set of what? Really all you seem to have is a concept of infinity. And you can only have this, as long as you do not define the set.

When we've been there ten thousand years,
Bright shining as the sun,
We've no less days to sing God's praise,
Than when we first begun.

An "actual infinite", no?

P.S.  Here's Wikipedia's article on Cantor's proof:

https://en.wikipedia.org/wiki/Cantor%27s...l_argument
[/quote]

No... that is what is referred to as a potential infinity.

[Jehanne]

No... that is what is referred to as a potential infinity.

Mathematicians (and, by extension, physicists) do not make that distinction:

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

There is no distinction between actual and potential infinity found in modern mathematics. Instead, infinite sets are assumed to exist in the axiomatic approach of the Zermelo–Fraenkel set theory.

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

[RoadRunner79]

No... that is what is referred to as a potential infinity.

Mathematicians (and, by extension, physicists) do not make that distinction:

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

There is no distinction between actual and potential infinity found in modern mathematics. Instead, infinite sets are assumed to exist in the axiomatic approach of the Zermelo–Fraenkel set theory.

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

You started by trying to make the distinction in calling it an actual infinity!  Now you are trying to slip away from it?

Have you figured out what your set of infinity is yet (defined it)?
And if you can apply infinity without any distinction are you really describing or saying anything at all?

[Jehanne]

Mathematicians (and, by extension, physicists) do not make that distinction:




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

You started by trying to make the distinction in calling it an actual infinity!  Now you are trying to slip away from it?

Have you figured out what your set of infinity is yet (defined it)?
And if you can apply infinity without any distinction are you really describing or saying anything at all?

Here it is:

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:



In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

https://en.wikipedia.org/wiki/Zermelo%E2...f_infinity
https://en.wikipedia.org/wiki/Axiom_of_infinity

[RoadRunner79]

You started by trying to make the distinction in calling it an actual infinity!  Now you are trying to slip away from it?

Have you figured out what your set of infinity is yet (defined it)?
And if you can apply infinity without any distinction are you really describing or saying anything at all?

Here it is:

In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:



In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.

https://en.wikipedia.org/wiki/Zermelo%E2...f_infinity
https://en.wikipedia.org/wiki/Axiom_of_infinity

What I am asking, is not to define what infinity is, nor to prove it as an abstract theory.  But when you are talking about two points, attributing infinity between those two points, what are you describing here?  Infinity of what?

It cannot be related to distance or a physical thing with dimensions, this would lead to a contradiction.  I don't think you can define what is infinite, without making in then finite.  You can potentially make smaller and smaller fractions, but each one will represent a finite amount, and will never get you an infinite result.

[Jehanne]

Here it is:



https://en.wikipedia.org/wiki/Zermelo%E2...f_infinity
https://en.wikipedia.org/wiki/Axiom_of_infinity

What I am asking, is not to define what infinity is, nor to prove it as an abstract theory.  But when you are talking about two points, attributing infinity between those two points, what are you describing here?  Infinity of what?

It cannot be related to distance or a physical thing with dimensions, this would lead to a contradiction.  I don't think you can define what is infinite, without making in then finite.  You can potentially make smaller and smaller fractions, but each one will represent a finite amount, and will never get you an infinite result.

The set of Natural Numbers is an (countably) infinite set, an actual infinite of numbers, even if we cannot "count" them.  So, too, are the set of numbers between any 2 numbers of the real number line, which would be an uncountable infinite set.  I realize that no one can "count" those numbers, just as no one can enumerate the number of past events, if, in fact, the Cosmos is without a beginning.  I am just claiming that "actual infinities" of physical things may exist, just as they do in transfinite arithmetic.  If space is infinitely divisible, and, time, too, then those are actual infinities of things; if not, then they are finite.  You (and, Craig) seem to be conflating the act of enumerating a set with the intrinsic membership of that set!  Again, what is the cardinality of the set of "future praises" in Heaven?!  Don't say that it is a "potential infinite"; I am not asking you to count it!  I am asking you what its cardinality is?!  Now, if you are going to insist on sets being "potentially infinite", then please define what the cardinality of a "potentially infinite set" is!  And, then, answer the question as to if some potentially infinite sets are bigger than others.

[RoadRunner79]

What I am asking, is not to define what infinity is, nor to prove it as an abstract theory.  But when you are talking about two points, attributing infinity between those two points, what are you describing here?  Infinity of what?

It cannot be related to distance or a physical thing with dimensions, this would lead to a contradiction.  I don't think you can define what is infinite, without making in then finite.  You can potentially make smaller and smaller fractions, but each one will represent a finite amount, and will never get you an infinite result.

The set of Natural Numbers is an (countably) infinite set, an actual infinite of numbers, even if we cannot "count" them.  So, too, are the set of numbers between any 2 numbers of the real number line, which would be an uncountable infinite set.  I realize that no one can "count" those numbers, just as no one can enumerate the number of past events, if, in fact, the Cosmos is without a beginning.  I am just claiming that "actual infinities" of physical things may exist, just as they do in transfinite arithmetic.  If space is infinitely divisible, and, time, too, then those are actual infinities of things; if not, then they are finite.  You (and, Craig) seem to be conflating the act of enumerating a set with the intrinsic membership of that set!  Again, what is the cardinality of the set of "future praises" in Heaven?!  Don't say that it is a "potential infinite"; I am not asking you to count it!  I am asking you what its cardinality is?!  Now, if you are going to insist on sets on being "potentially infinite", then please define what the cardinality of a "potentially infinite set" is!  And, then, answer the question as to if some potentially infinite sets are bigger than others.

So you have a set of numbers that do not represent anything?  Then they are just numbers, that do not correspond to the two points in any way.

And what you are describing is a potential infinite, because per your wikipedia page

This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

Do you agree, that even if you can potentially divide something infinity, that your results each time are a finite number?

[Jehanne]

The set of Natural Numbers is an (countably) infinite set, an actual infinite of numbers, even if we cannot "count" them.  So, too, are the set of numbers between any 2 numbers of the real number line, which would be an uncountable infinite set.  I realize that no one can "count" those numbers, just as no one can enumerate the number of past events, if, in fact, the Cosmos is without a beginning.  I am just claiming that "actual infinities" of physical things may exist, just as they do in transfinite arithmetic.  If space is infinitely divisible, and, time, too, then those are actual infinities of things; if not, then they are finite.  You (and, Craig) seem to be conflating the act of enumerating a set with the intrinsic membership of that set!  Again, what is the cardinality of the set of "future praises" in Heaven?!  Don't say that it is a "potential infinite"; I am not asking you to count it!  I am asking you what its cardinality is?!  Now, if you are going to insist on sets on being "potentially infinite", then please define what the cardinality of a "potentially infinite set" is!  And, then, answer the question as to if some potentially infinite sets are bigger than others.

So you have a set of numbers that do not represent anything?  Then they are just numbers, that do not correspond to the two points in any way.

And what you are describing is a potential infinite, because per your wikipedia page

This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

Do you agree, that even if you can potentially divide something infinity, that your results each time are a finite number?

No, I am saying that the sum of those finite numbers (their ordinality) is infinite. But, once again, please answer my question, "Are some potential infinities bigger than others?"

[RoadRunner79]

So you have a set of numbers that do not represent anything?  Then they are just numbers, that do not correspond to the two points in any way.

And what you are describing is a potential infinite, because per your wikipedia page

Do you agree, that even if you can potentially divide something infinity, that your results each time are a finite number?

No, I am saying that the sum of those finite numbers (their ordinality) is infinite.  But, once again, please answer my question, "Are some potential infinities bigger than others?"

I agree, that you can make up fractions potentially forever.  However what they represent is still part of a finite thing.  You are talking about the process.
I do think, that the more loosely define your set, then a infinite multi-dimensional array would be technically larger than and infinite single dimension array (such as used in addition).  

How do you apply this to the topic?
Its seems, that you are talking about abstractions, not in any way reference to the two points of the OP.  How do you connect these?

Do you see the logical contradiction with having an actual completed set, that is by definition never complete?