Certain people can't ever get it right

[Physics Guy]

We are contending, then, that an actual infinite cannot exist in the real world. It is usually alleged that this sort of argument has been invalidated by Cantor’s work on the actual infinite and by subsequent developments in set theory. But this allegation seriously misconstrues the nature of both Cantor’s system and modern set theory, for our argument does not contradict a single tenet of either. The reason is this: Cantor’s system and set theory are concerned exclusively with the mathematical world, whereas our argument concerns the real world.

This just sounds really dumb.

If there really were an infinite number of red and blue books then it would be like that, because books may be real or hypothetical but numbers are numbers. The two-ness of two apples is the same as the two-ness of two unicorns.

So I don’t see how it makes any difference whether the books are real or mathematical. If Craig thinks it does make a difference then as far as I’m concerned he’s only digging himself deeper into the hole.

Let me fix Craig’s typo:

What I shall argue is that while the actual infinite may be a fruitful and consistent concept in the mathematical realm, it cannot be translated from the mathematical world into the real world, for this would involve counter-intuitive surprises for me and I am too ignorant of natural science to realize that the real world is often counter-intuitive to humans.

[Lem]

For instance, if an actual infinite could exist in reality, then we could have a library with an actually infinite collection of books on its shelves. Remember that we are talking not about a potentially infinite number of books, but about a completed totality of definite and distinct books that actually exist simultaneously in time and space on these library shelves. Suppose further that there were only two colors of books, black and red, and every other book as the same color. We would probably not balk if we were told that the number of black books and the number of red books is the same. But would we believe someone who told us that the number of red books in the library is the same as the number of red books plus the number of black books?

Craig thinks infinity is a number?

This just sounds really dumb.

It does.

[malkie]

I kind of like how my favourite poet puts it:

The universe may
be as great as they say.
But it wouldn't be missed
if it didn't exist.

[Gadianton]

For we C students:

They see the fact that an infinite set can be put into one-to-one correspondence with one of its own proper subsets as one of the defining characteristics of an infinite set, not an absurdity. Say that set C is a proper subset of A just in case every element of C is an element of A while A has some element that is not an element of C. In finite sets, but not necessarily in infinite sets, when set B is a proper subset of A, B is smaller than A. But this doesn’t hold for infinite sets—as above where B is the set of squares of natural numbers and A is the set of all natural numbers.

https://plato.stanford.edu/entries/cosm ... poActuInfi

If it's a venial sin to describe the universe with an idea like infinity that makes intuitive sense in math but not everyday examples in the real world, then is it a mortal sin to describe the universe with an idea like the square root of negative one, that doesn't even make intuitive sense in math?

[Philo Sofee]

Infinity is not a number. Craig has been told this innumerable times. It is not a number, a quantity, it is a quality, which is why it is non-denumerable, meaning you can't count up to it. You can't calculate to get to infinity anymore than you can give a mathematical equation of the quality of Virtue, or Patriotism, or Hope. Yet, those are very real qualities of a human life. One does not say because there are contradictions the qualities don't exist, unless one's name is William Lane Craig apparently. The precise reason contradictions occur with infinity is to this point. In reality, there is no contradiction.

With the concept of a finite thing, say a red rubber ball a little boy plays with that has a diameter of 6 inches, it's small enough he can play with it. It has finiteness. The very concept of limit, the idea that something is only so big or small, already demonstrates the opposite has to be, or we could not have the concept. With light there is shadow. This is in the field of space/time. Finite automatically implies infinity or as a quality finiteness would literally make no sense.

[Physics Guy]

I don't think the point here is about what does or does not count as "a number". Transfinite cardinalities are a generalisation of the concept of a number; they retain enough of the intuitive properties of numbers that it doesn't seem wrong to me to call them numbers, in a broad sense of "number".

The point is that just because infinity is strange doesn't mean it's impossible. A philosopher who argues baldly, "This seems strange to me, therefore it must be impossible," is not worth hearing. And what Craig is saying is no more than that.

I have to say I'm particularly irked by his pompous little preamble about how objections to his argument are based on misconstruals of modern set theory, because this is an implicit claim that Craig understands set theory and transfinite cardinality better than his critics do. Since he evidently imagines that his sophomoric distinction between real and mathematical worlds is a get-out-of-jail-free card for the logic of infinity, however, it is clear that Craig doesn't really understand these subjects at all. If all Craig did were to ramble about blue and red books I would let him plead guilty to the lesser charge of ignorance—nobody can know everything—but this attempt to bluster and snow by name-dropping "Cantor" and "modern set theory" is annoying.

I don't think Georg Cantor is even relevant here, in fact. Cantor's big contribution, as far as I know, was uncountable infinities, like the cardinality of the irrational numbers. Craig's red and blue books don't go any farther than mere countable infinity (the cardinality of whole numbers). I don't think it takes Cantor or modern set theory to deal with that.

[Edit: In my original post I mentioned as an example of an uncountable infinity the cardinality of the rational numbers. This wasn't just a typo but it was a mistake: I had forgotten my math. The rational numbers are countable. It is the irrational numbers that are not, and this was Cantor's big discovery.]

[Lem]

I don't think the point here is about what does or does not count as "a number". Transfinite cardinalities are a generalisation of the concept of a number; they retain enough of the intuitive properties of numbers that it doesn't seem wrong to me to call them numbers, in a broad sense of "number".

Would that Craig used the term "number" thusly.

The point is that just because infinity is strange doesn't mean it's impossible. A philosopher who argues baldly, "This seems strange to me, therefore it must be impossible," is not worth hearing. And what Craig is saying is no more than that.

There was a poster here several years ago who used to say he "knew" something was true or not based on his "gut" reaction, and then would argue from there. One of his last topics he argued in this manner was whether being transgender was a real thing. I kid you not. He was incensed when anyone would ask him to back up his "gut" with facts, implying that it was utterly ridiculous that an argument would include a need for him to justify his "gut."

I have to say I'm particularly irked by his pompous little preamble about how objections to his argument are based on misconstruals of modern set theory, because this is an implicit claim that Craig understands set theory and transfinite cardinality better than his critics do. Since he evidently imagines that his sophomoric distinction between real and mathematical worlds is a get-out-of-jail-free card for the logic of infinity, however, it is clear that Craig doesn't really understand these subjects at all.

Maybe his "gut" is talking to him.

[SeerOfProvo]

This just sounds really dumb.

That is the hazard of quoting just a few paragraphs of a larger and complicated work. A handful of snapshots can be misleading to the person who didn’t choose, line up, and take the shots themselves. The reality is Craig is marshalling arguments originating from the debates raging between scholars about the foundation of mathematics during the 20th century.

What Craig is articulating comes straight out of the so-called “intuitionist” school and really isn’t anything the likes of a L.E.J. Brouwer hasn’t already put out there, especially in his contentious debates with David Hilbert and others. In fact, even David Hilbert and John von Neumann were ready to sacrifice actual infinities and transfinite induction in the so-called “formalist” metamathematics of the Beweistheorie.

Craig isn’t just inventing this stuff in some dusty basement under the halls Talbot for an audience of seminary students. He is simply taking a minority position and making a defense of it so that his Cosmological argument has potential, we can’t begrudge him for doing that.

If there really were an infinite number of red and blue books then it would be like that, because books may be real or hypothetical but numbers are numbers. The two-ness of two apples is the same as the two-ness of two unicorns.

Don’t get me wrong, I think our modern notions of set theory are correct and true, if they are correct and true then they are by necessity, so it seems to me that our universe is going to conform to it. That being said, that includes a host of assumptions and it isn’t absurd to call each of those assumptions into question.

So I don’t see how it makes any difference whether the books are real or mathematical. If Craig thinks it does make a difference then as far as I’m concerned he’s only digging himself deeper into the hole.

Another underlying issue here is the reality of mathematical objects; Craig doesn’t think mathematical objects exist, but if they do exist they do so in an abstract manner that is causally inert with no spatiotemporal properties. Now Craig happens to be an anti-realist in this sense, but he doesn’t have to yield anything by simply pointing out that there is a difference between a mathematical object which is entirely conceptual in nature and a book with physical dimensions and dispositional properties. I mean a mathematical object and a physical book are far more different in nature than an apple and unicorn could ever be, is it really digging himself a hole to ask why we think one is going to behave just like the other?

Maybe you agree with Craig and even go so far to say that Mathematics are invented (not discovered); a human language used to describe the world. Mathematical objects are merely these conjectures we’ve fashioned much like our human alphabet that are extremely useful, but much like unicorns, just products of the human psyche.In this instance Craig can still ask the same question, on what basis do we think physical books can be treated identically to these concepts of a formal language?

Like I said before, I strongly believe Craig is wrong, but he can be wrong in interesting ways.

Craig thinks infinity is a number?

He uses ω = א (subscript 0) as a number just like everyone else does.

Infinity is not a number.

Slipping into an informal use of infinity to make a point isn’t an error on Craig’s part, it happens a lot of mathematical texts.

Craig has been told this innumerable times.

By who, exactly?

You can't calculate to get to infinity anymore than you can...

Yes you can (in the sense of a maximal ordinal line), it's called transfinite induction (and recursion*) and there are rules and operations for it. All of this is assumed by both Craig and the people he is disagreeing with in the text, the mechanics are not disputed at all.

[Physics Guy]

I still don't see anything interesting in Craig's red-and-blue-books argument. Mentioning real and mathematical worlds is just a red herring because the only features of the books that his argument uses are that they are of two distinct kinds and that they can be counted. Both are mathematical properties which do not depend on anything physical. He's not saying we couldn't build a shelf that would hold all those books; he's not talking about the physical possibility of printing infinite numbers of books, although for this we have scriptural warrant in Ecclesiastes 12:12.

Craig objects purely to a mathematical property of an infinite number of books. His argument is of the same logical form as saying that we cannot have nine books because if we did, why, the number of books divided by three would be three and that is absurd. In Craig's case the claimed absurdity is that the cardinality of an infinite set is the same as that of of one of its subsets. The only sense in which that is absurd is that it implies that the set is infinite.

So Craig's argument is pure question-begging: we cannot have an infinite number of real things because if we did, why, we'd have an infinite number of them and that is absurd.

Either that or he's making a silly equivocation on "number", referring first to the number of books in a generalised sense of number that includes infinity but then invoking our intuitions for finite numbers without drawing attention to the fact that he has shifted his definition of "number" between one phrase and the next.

Neither of these arguments interests me.

[Moksha]

Would an infinite number of books achieve critical mass?