How about SET-SIZE(N) = oo ? Can we have some Terminology back??

[Graham Cooper]

Of course, another i-word is significant to this thread: "infinity."
We see
that Herc rejects the distinction between countable and uncountable
that's found in standard set theories like ZFC, and instead proposes
that all infinite sets have set size "infinity." But the others argue
that
the word "infinity" shouldn't be used this way -- one can say that a
set
is _infinite_, but not that it has set size "infinity."
I've seen previous threads debate the use of that word "infinity," and
I
have a question which I've yet to see answered satisfactorily in any
of
the previous "infinity" threads: What exactly does the word "infinity"
mean in the axiom name "Axiom of Infinity"? An answer to this might
help settle the debate.

Since me and Jesse skimmed the topic of PreCantorian Infinity,
I might add the question, "Is AOI consistent with Peano Axioms?"

or in this case, PEANO + "e" + AOI

or by the time you do that you may as well be talking Naive Set Theory
or ZFC again which makes the question "how long is an infinite piece
of string?"


Herc

[DonH]

Of course, another i-word is significant to this thread: "infinity."
We see
that Herc rejects the distinction between countable and uncountable
that's found in standard set theories like ZFC, and instead proposes
that all infinite sets have set size "infinity." But the others argue
that
the word "infinity" shouldn't be used this way -- one can say that a
set
is _infinite_, but not that it has set size "infinity."
I've seen previous threads debate the use of that word "infinity," and
I
have a question which I've yet to see answered satisfactorily in any
of
the previous "infinity" threads: What exactly does the word "infinity"
mean in the axiom name "Axiom of Infinity"? An answer to this might
help settle the debate.

Since me and Jesse skimmed the topic of PreCantorian Infinity,
I might add the question, "Is AOI consistent with Peano Axioms?"

or in this case, PEANO + "e" + AOI

or by the time you do that you may as well be talking Naive Set Theory
or ZFC again which makes the question "how long is an infinite piece
of string?"


Herc

# "How long is an infinite piece of string?"
This can be slightly re-stated to provide its own answer -
"How long is an in-finite (non-ending) piece of string?"
Naive? Or, merely definitional?