Re: Anti-diagonalist page

Ullrich---
The author of the text I was refuting was not denying the
existence of an uncountable set.
---

I asumed he was. Note he also puts scare quotes around 'sets'.

---
Anyway, the point is that Slater is not objecting to
the truth of the fact which mathematicians express
by saying that the cardinality of the reals is greater
than the cardinality of the naturals.
---

On the assumption that "the cardinality of the reals is greater than
the cardinality of the naturals" requires that infinite sets exist, I
assume he would be objecting to this. I'm basing this on more of
Slater's writing than the brief quotation I gave on this list. For
example, his paper 'Grammar and Sets' which is to appear in the
*Australasian Journal of Philosophy* next March. Apologies if this was
given out of context.

---
He's objecting to _saying_ that this entails that there are "infinite
numbers" of various sorts. Which is silly, because _that_ is just a
matter of definition.
---

Even if I have misunderstood his point, there is still the objection
that you can't define something ('number') that already has a meaning.
One may agree with "the cardinality of the reals is greater than the
cardinality of the naturals" but disagree that "cardinality" means
"number" in the ordinary sense. That is also W.'s point.

---
To give another, perhaps better reply to this: If we insist
on making an analogy with unicorns the analogy would be
this: Slater says that yes, the proof of the existence
of horses with horns _is_ valid, but he objects to inferring
from that that unicorns exist. Which is simply silly.
Once we have accepted that horses with horns do exist
_then_ the existence of unicorns _is_ just a matter of
definition.
---

Agree, and apologies if I misunderstood your point here. I was dealing
with three long replies at once.

---
Lemme try to summarize for you. Slater says yes, the proof
that there is no bijection from the naturals to the reals
is valid, but he insists that this is not evidence for the
existence of "infinite numbers" of various sizes. To make
this coherent we need to write "cardinal" for "infinite number".
If we do that then yes, the theorem does prove that there
are infinite numbers of various sizes, by definition of
"cardinality". Insisting otherwise is just silly.
---

No, it proves that there are cardinals of various sizes.

.

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