[FOM] Cantor and "absolute infinity"

Harvey Friedman friedman at math.ohio-state.edu
Sat Feb 18 23:35:47 EST 2006

On 2/18/06 6:09 PM, "Arnon Avron" <aa at tau.ac.il> wrote:

> ...the
> obvious question would have been (and was later):
> what is the criterion that distinguishes between "absolutely
> infinite" collections and non-absolute infinite collections
> (= infinite "sets")?

>... Did he have any characterization of this "set",
> except as the  set of all sets which are subsets of A
> (a clear application of unrestricted comprehension)?
> ...Cantor [needed] to give
> a convincing argument why P(N) is a set
> and not an absolutely infinite collection .... I cant see what
> this argument could have been. In fact, I cant see what
> criterion Cantor could have had for telling what collections
> are "absolutely infinite" (and according to which
> P(N) is not "absolutely infinite") except to ban collections/sets
> the assumption of which leads to contradictions...

This issue is easily resolved if you look at set theory as an extension of
finite set theory. 

Specifically, HF = the *infinite set* of all hereditarily finite sets, is
the standard model of (pure) finite set theory.

For HF, the "absolute infinities", or "proper classes" are just the
*infinite subsets* of HF. The power set demonstrably holds in HF.

In clear analogy, V = the *proper class* of all sets, is the standard model
of (pure) set theory.

For V, the "absolute infinities", or "proper classes" are just what we now
call the *proper subclasses* of V. The power set holds in V --- in analogy
to the HF case.

The situations are entirely analogous.

> ... Cantor's talks about absolute infinity were caused
> by Cantor's paradox, a paradox that occurs to every good student who
> learn for the first time Cantor's theorem (that the powerset of
> A has bigger cardinality than that of A).

The analogy to "Cantor's paradox" in finite set theory is that "the set of
all hereditarily finite sets is not an hereditarily finite set".

>... here come the 
> vague idea of "absolute infinity".

"here comes the vague idea of "infinite set (of hereditarily finite sets)"
". Not so vague for most of us.

> The upshot is: Hilbert, Dedkind, Weyl etc were not so stupid
> after all, when they saw that the paradoxes are threatening
> Cantor's theory too, and not only Frege's.

I wouldn't call these people "stupid", but if they understood the proper
analogy with finite set theory, they would see that the paradoxes do not
threaten Cantor's theory at all.

>What is problematic
> for both is the comprehension principle.

In finite set theory, the comprehension principle is not problematic as it
follows from induction. Same with power set. Set up properly, every axiom of
finite set theory follows by induction (over extensionality plus adjunction:
x union {y} exists. Induction formulated with x union {y}, of course).

>Unfortunately, this
> principle leads to contradictions.

Only when stated so generally that it has no analogue in finite set theory
(in fact, refutable in finite set theory). The obvious form of comprehension
in finite set theory is separation, which apparently does not lead to any

> I think that Hilbert, Dedkind, Weyl etc deserve more credit than
> thinking that if only they knew Cantor's hidden views they would never
> have seen problems in set theory.

If one wants to consider alternate historical paths, I am confident that
these three would have been very struck by the strong analogy with finite
set theory and what I have outlined above. This may well have been decisive
for some or even all of them.

>I cant 
> imagine any of them thinking "Aha, Cantor
> meant only sets that are not absolutely infinite! Why has nobody
> told me before, and how come I could not have understood it myself?").

They would have said "Aha, Cantorian set theory is the OBVIOUS extension of
finite set theory to the infinite, and this explains a great deal".

Harvey Friedman

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