I am a little confused with your definition of an infinite set - if S** is a finite proper subset of S* then N* < N must hold? The text you've quoted though reads to me as if he is saying that infinite sets do not have a cardinality and are not well defined?Consul wrote: ↑July 11th, 2019, 1:51 pmWe know how many items there are in an infinite set:infinitely many. Of course, this answer is indeterminate insofar asinfiniteis not a number; and the question "What's the number of members of an infinite set?" cannot be answered in terms of anyfinitenumber. Can it be answered in terms of sometransfinitenumber? 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 literallynumberless, orbeyond number."

(p. 143)

"[A]lthough infinitude is surely a quantitative property, it is not adeterminatequantity. To call something infinite is to saysomethingabout its number or size, but it is not toassign a specific valueto 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 hadnothing to dowith 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 nota 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.

I think there is a general rule here: 'To be X, you must start X'. So X could be counting, reciting, spinning, oscillating and ultimately existing. The initial state of all these activities determines the subsequent states. If there is no initial state, there are no subsequent states.Consul wrote: ↑July 11th, 2019, 1:51 pmI think this argument is unsound, because counting is arguably an activityrequiring 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 isreciting 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.

The number on its own is well defined. It is the whole infinite series containing it that I argue is undefined.Consul wrote: ↑July 11th, 2019, 1:51 pm

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."