Re: etc

Andy Smith wrote:
All right, commenting from a position of ignorance as usual:

My mental view of the development of mathematics (as exemplified by
numbers) was :

1) natural numbers
2) rationals
3) irrationals
4) negative numbers and zero
5) transcendentals
6) complex numbers
7) reals covering the continuum (uncomputable reals)

Possibly I have the historical order wrong.

If we are discussing history, then I think the positive reals have to go
ealier--probably at #3. Maybe the positive transcendentals go before
zero and negatives; I'm not sure. Of course, the transcendentals are a
subset of the irrationals.

But when you say above
The axioms prove there is an ordered pair of real numbers having
the required property.
was surely not true until such time as the axioms of real number
arithmetic were extended to encompass a more general object (not a real
number) having the property that its "square" (under some definition of
multiplication for such not-a-number entity) = -1, and that all of that
was shown to be free of contradictions (maybe).

MoeBlee wasn't discussing history. He was discussing our present view.
In particular, the "axioms" he is referring to are those of ZFC, so this
is fairly recent.

Generally, all of this historical development involved a development of
awareness of the concept of "algebra" to describe a given structure -
so complex numbers have a definition of complex multiplication,
conjugation; modulo arithmetic has its own set of rules, etc. (As i
understand it you define a "field" and axioms, and rules of inference
that follow standard logic; I'm not quite sure how you show that
everything is consistent in the sense of showing that you can't prove A
and not A in a given system, but that is a different conversation, but I
am sure that you do).

Usually, we prove consistency by producing a model for the axioms. For
example, if we write down the axioms for groups, we show an example of a
group. This tells us the axioms are talking about something. We do the
same for number systems.

So my comment re infinite sets was just that, I bet, in historical
terms, the concept of a set "a many to be considered as a whole", was
originally finite, and that the axioms of ZFC implicitly or explicitly
extended that to allow the concept of an infinite set as a "finished
thing".

Clearly, the axioms came after the concepts. So, I'm not sure what you
mean. However, math has been dealing with infinite things for a very
long time.

So it was in that sense that I suggested that it was a matter of
choice that one might "allow" the concept of an infinite set, extending
mathematics in a comparable fashion to that involved in the extension
from the real line to the complex plane.

But, what does it mean to not allow infinite sets? I really don't know.
It would seem that you would have to throw out pretty much all of
mathematics. Of course, you can avoid talking about them, and go back to
the way the math was presented in previous centuries. But, if we are
going to use sets as our language, then how can we avoid infinite sets?

Since I am here, am I right in thinking that there is no mathematical
(or physical, bearing in mind that eigenstates of wave functions relate
to integer constraints) process that would allow the construction, even
in principle, of a non-computational number? Even though the
computational numbers on interval [0,1] have measure 0 and the
non-computational numbers have measure 1, one cannot define a pin,
randomly or otherwise, sharp enough to identify a non-computational
number in [0,1]? And that the sets of computational and
non-computational reals are disjoint - there can be no relationship
between the two subsets of the reals - you can't specify a computational
real in terms of some function of non-computational reals, and
vice-versa?

Not sure what most of that means, but to answer your question "am I
right", I think the answer is no. Here is a simple thought experiment:
Write a decimal point. Now, toss a coin. If it is heads, write "1". If
it is tails, write "0". Repeat tossing the coin forever. We can
interpret what you have written as a binary number. The probability that
the number you write is a computable real is zero.

--
David Marcus
.

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