What you are presuming is that actual infinities really exist. Actual infinity, however is merely an abstract notion , or, if you like, a fiction in the realm of the philosophy of mathematics which proposes that mathematical objects like , say the infinite sequence of negative numbers you refer to above can form a complete totality or "set", I.e. a given object that is a true actual infinity. Actual infinities, though, do not exist, they are not realities. To understand why this is you should google up and read the mathematician David Hibert's famous thought experiment , "Hibert's Hotel". — John Gould
Thanks for your comments.
I disagree with you on three points, summarized as follows.
1.
Even if I were using actual infinity, so what? After all, actual infinity is no weirder than an all-knowing, all-powerful, benevolent uncaused cause.
2. However, I am NOT using actual infinity. Your understanding of potential versus actual infinity is different than Aristotle's. My model "..., -3, -2, -1" only uses POTENTIAL infinity as defined by Aristotle. I will expound on this point in a moment.
3. Hilbert's Hotel (HH) doesn't apply. You are correct that HH assumes actual infinity. But my example only requires potential infinity. I don't need all the numbers (or rooms) to exist all at once. I only need that given one, I can identify the next. I never assume I have them all existing at once. That's potential infinity.
Here is more detail, especially on point #2.
1. For the moment let me grant your (false) premise. Say I did need actual infinity (which I remind you I don't). So what? Craig wants us to conclude that there must be an uncaused cause, which he calls God. What if I call it absoulte infinity? I can't conceive of a worldview that would
grant divinity but not infinity.
Cantor himself thought that his Absolute infinity was God. But Cantor's Absolute infinity is a lot bigger than the infinity of the natural numbers.
2. But #1 is irrelevant, since I don't use actual infinity, only potential. Let me outline the concept as seen by Aristotle. He said that we all have an intuition of the natural numbers 0, 1, 2, 3, 4, 5, ... Now the "dot dot dot" means something specific as defined by the
Peano axioms:
*
Inductive axiom: Given a number n, there is a number n' called the "successor" of n.
Another notion for the successor of n is n + 1. So if 0 exists, then 1 does. If 1 exists then 2 does. If 2 exists then 3 does.
So if you want to know, does 43242342 exist? Then you can recursively drill down all the way back to 0, the base of the induction, and you can show that any particular number exists.
There is never any claim that we have all of them together all at once. We can imagine they don't come into existence till we need them. All I need is n + 1 given n. If I need a million, I make a million. I never have them all at once.
That is exactly
Aristotle's definition of potential infinity. In the following quote, Aristotle is speaking of the endless regress 1/2, 1/4, 1/8, 1/16, etc. He says:
"For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately."
— Metaphysics, book 9, chapter 6.
Now the sequence 1/2, 1/4, 1/8, ... may be identified with the the reverse sequence I gave earlier, ..., -3, -2, -1 by the simple mathematical trick of taking the base-2 logarithm of each fraction to get the corresponding negative integer.
1/2 maps to -1, and 1/4 maps to -2, and so forth. So these two examples are really the same example in different forms.
Aristotle calls this
potential infinity. Does 5 exist? Yes, if 4 exists. So we can prove that any number n exists. But we can't say that ALL the counting numbers exist all at the same time. That would be ACTUAL infinity.
It was the genius of Cantor to take the huge conceptual leap and say, What if we allow actual infinity into math? That was his brilliant history-changing leap of the imagination.
Cantor's insight was to write the following notation:
{0, 1, 2, 3, ...}
The braces symbolize the COMPLETED SET of natural numbers. The inductive axiom gives us 1 (given 0) then 2, then 3, and so forth.
The
Axiom of Infinity says that there is a set containing all the natural numbers. That's actual infinity.
We can summarize all this with a table. I apologize that I could not make the right column line up no matter what I did with tabs and spaces. Advice appreciated.
Potential infinity Actual infinity
Axiom of induction Axiom of Infinity
Peano Cantor
0, 1, 2, 3, ... {0, 1, 2, 3, ...}
n+1 given n All of them at once
Negative integers Hilbert Hotel
I hope this is helpful.
3. HH is just a popularized visualization of the fact that an infinite set may be placed into bijection with one of its proper subsets. In fact this can be taken as the definining property of infinite sets.
You are right that HH does assume actual infinity. But we do NOT need actual infinity to define the natural numbers 1, 2, 3, ... in their usual order, or ..., -3, -2, -1 in their reverse order, or 1/2, 1/4, 1/8, ... if you use base 2 exponentials and logarithms.
I don't need the strength of the axiom of infinity to give my example. Only the Peano axioms, which define potential infinity. Given n I need n + 1. I never need to complete the process. I only need to take the next step. So Hilbert's Hotel is not relevant here.