devans99 wrote: ↑July 11th, 2019, 1:05 am

Consul wrote: ↑July 10th, 2019, 6:01 pm

The general point is that even if there is no such thing as the number of things in an actual infinity of things, it doesn't follow that there isn't any actual infinity of things.

For example, I think you must agree that the set {1,2,3} is 'more defined' than {1,2,3,4,...} - we know how many members there are in the first. So the fact that finite set is 'more defined' than the infinite set can be taken as indicating that the infinite set is not fully defined - IE UNDEFINED. I think maths defines sets inconsistently - you can have a fully finite defined set like {1,2,3} but it also allows just the 'selection criteria' for a set to count as a set (by selection criteria, I mean for example 'all negative integers'). In my book, the 'selection criteria' is different from the set itself and the selection criteria does not completely define a set.

We know how many items there are in an infinite set:

*infinitely many*. Of course, this answer is indeterminate insofar as

*infinite* is not a number; and the question "What's the number of members of an infinite set?" cannot be answered in terms of any

*finite* number. Can it be answered in terms of some

*transfinite* number? Yes, but if the transfinite cardinal numbers (Aleph-numbers) are just pseudonumbers, then the only possible answer is the following one:

*"Although there are actual instances of infinitude, there are no infinite numbers. When, for example, there are infinitely many A’s, the A’s do not instantiate some specially large number; rather, the A’s are literally ***numberless**, or **beyond number**."
(p. 143)

*"[A]lthough infinitude is surely a quantitative property, it is not a ***determinate** quantity. To call something infinite is to say **something** about its number or size, but it is not to **assign a specific value** to its number or size. Rather, an ascription of infinitude is essentially negative: it says, roughly, that any assigned number or size is insufficient."
(p. 147)

*"The thesis that infinity is not a number naturally invites the speculation that an infinite thing cannot even be said to be greater than a finite thing, because ‘greater’ is a relation that only makes sense when applied to numbers.*

The above discussion, however, hints at the reply to this concern. If infinitude simply had **nothing to do** with number – if an ascription of infinitude said nothing about size or numerousness – then indeed an infinite thing would not thereby be larger than a finite thing. But that is not the view suggested here. The view is that infinity is not **a specific numerical value** (nor is Aleph_0 or any of the other alleged infinite numbers). An infinite thing may definitely exceed the finite, while bearing no determinate quantitative relation to another infinite.

More precisely, the infinite may exceed the finite by having finite parts that are greater than any chosen finite object."
(pp. 147-8)

(Huemer, Michael.

*Approaching Infinity.* New York: Palgrave Macmillan, 2016.)

So, an infinite set can be defined as a set S which is such that for all subsets S* with a finite cardinality N there is some subset S** with a finite cardinality N* such that N* > N.

devans99 wrote: ↑July 11th, 2019, 1:05 am

Consul wrote: ↑July 10th, 2019, 6:01 pm

But that's just a reiteration of your

*premise*—which is false.

Well, try it with another sort of infinite regress - an immortal alien who has always been counting - he can't be on any finite number (because then he'd be mortal) and he can't be on infinity (because its impossible to count to infinity). So that rules out ALL the numbers - he can only be on UNDEFINED (because he never started count) - infinite regresses are impossible.

I think this argument is unsound, because counting is arguably an activity

*requiring a beginning or start*, such that it's not possible for an immortal alien to have always been counting, in the sense that there is no time in the past when he isn't counting.

What doesn't require a beginning or start is

*reciting some number or other (e.g. per day)*—without any particular numerical order as in the case of counting. So your immortal alien could always have been reciting some number or other (per day) without this being a case of counting.

devans99 wrote: ↑July 11th, 2019, 1:05 am

Consul wrote: ↑July 10th, 2019, 6:01 pm

Can you tell me e.g. what's "undefined" about the number –256?

-256 is fully defined.

This is in contradiction with what you wrote in a previous post, since -256 is part of an infinite series lacking a first element:

*"To reiterate my early argument - if there is no first element in a regress (IE the bit is UNDEFINED rather than 0 or 1), then the second element is UNDEFINED and by induction the whole regress is UNDEFINED."*