Re: An uncountable countable set


Tony Orlow wrote:
MoeBlee wrote:
Tony Orlow wrote:
If the naturals are not a subset of the reals in ZFC and NBG, then those
theories are even more screwed up than they already seemed.

With either of the usual constructions of the reals (Dedekind cuts or
equivalence classes of Cauchy sequences), both the system of naturals
and the system of rationals are isomorphically embedded in the system
of reals. Mathematicians usually speak of the system of natural numbers
and the system of rational numbers as subsystems of the the system of
reals. Strictly speaking, that is incorrect, but it is harmless given
the isomorphism.

Harmless, even though incorrect? What makes it incorrect, if not
inconsistency? Doesn't inconsistency cause a problem?

Now you're being captious. The informality is harmless, as I said,
because we are within ISOMORPHISM. It has nothing to do with the
consistency of the theory. It's only a matter of INFORMAL convenience
to not have to say each time "the system that is isomorphically
embedded" but instead speak directly of the system as if it were a
subsystem, since, to WITHIN ISOMORPHISM, it is a subsystem. This kind
of informality is common throughout mathematics and is harmless.

I used y in the equation (or statement of inequality) and then specified
in parentheses at the end, with a space, which condition for y made that
statement true. I didn't think that was tooooo confusing.

It was unclear enough that I had to check with you what you meant. It's
not a big deal, but you could acheive clarity by being more explicit.

'number of any real sort' is not a defined predicate of set theory. w
is not in the standard ordering of the reals. That does not make w an
undefined object.

My experience is that asking amthemticians for a definition of "number"
results in.....nothing.

Because it's more a philosophical issue or an issue of terminology
outside the system. The purpose of set theory is not to address the
question of what is and is not a number. Rather, among the purposes of
set theory is to axiomatize and construct the various number systems
that are of interest.

you don't see scientists accepting transfinitology either.

Of course, you have a survey of scientists to support your claim.

Uh, yes, right here. Why don't you survey thise that object, regarding
what they do?

That's quite an unscientific survey method of yours.

You have no coherent system of definitions at all. It's not the job of
set theory to define objects that obey the whims of your informal
notions.

It's the job of mathematicians to work with numbers.

Numbers are among the primary concern of mathematics. Set theory
axiomatizes and constructs number systems.

The selection of any unit is done by simply choosing a point separate
from the origin. When it comes to division using infinite values, one
translates the geometric definition into symbolic form and applies
induction formulaically.

"Translates the geometric definition into symolic form and applies
induction formulaically" is no less doubletalk than "Coordinates the
numeric form into geometric postulates and applies the recursive
definition metrically."


Is that a question?

Is that a rhetorical question?

MoeBlee

.

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