[FOM] The Omega Express (Set Theory and Ordinary Language)
Dean.Buckner at btopenworld.com
Tue Dec 31 07:25:04 EST 2002
Thanks for the helpful replies to my previous posting.
Let me clarify my problem. We are comfortable with the ordinary language
predicate "- is a natural number". We prove statements that seem to
quantify over objects that satisfy this predicate. Thus we grasp the
concept that it signifies, in order to do which you would think there need
not be anything that satisfies the concept. (We can e.g. define the concept
"is a unicorn" as "is a horse with a horn", even though there are no
Yet in ZF, we have to assume the existence of a number of different objects
in order to define what is apparently the same concept. We have to posit
(a) an empty set (b) rules that guarantee the existence of infinitely many
objects, from the existence of the empty set (c) the existence of a set
containing all these objects.
We need to allow quantification over the infinitely many objects guaranteed
by (a) and (b). To prove something of any number x, we have prove something
of the form "for any x, such that x is a number ..." which in turn requires
taking an arbitrary object a, such that the object satisfies the concept "-
is a natural number". But we can't do that unless we pick out the right
It is like picking a station that a train will pass through, given only the
existence of a station from which the train starts, and the existence of
stations, every one of which has station before it, and a station after it.
Given this alone, we can't guarantee that a station we select at random will
be such that the train passes through it. We need first to postulate the
existence of a set of stations that the train passes through, in order to do
But the analogue of positing the set of stations in ZF, is postulating the
existence of a set of natural numbers, i.e. the Axiom of Infinity. We can't
define the concept "- is a natural number" in ZF without postulating the
existence of an infinite set of numbers.
Yet we can define the same concept (or one that looks very similar ) in
ordinary language. With our finite minds we are able to grasp something
whose definition requires an infinitude of things. That's not necessarily a
contradiction, but it seems odd to me.
That's my problem.
I think I see a solution (involving the fact that, in ordinary language, we
do not quantify over "numbers" at all), but before that, does this set the
problem up in the right way? Is it in fact true that we can't define the
concept "is a number" in ZF without AxInf?
Wishing a Very Happy New Year to all those involved in FOM.
PS this "train" of thought was set off by Jeremy Clark's remark that "no
collection of conditions specifiable in first-order sentences will capture
finite linear-orderings, as a result of the compactness theorem".
The focuses the puzzle. You've already got a bunch of objects 0, 1, 2, 3
and so on without AxInf. Every single object that we need for set theory
already exists at this point. You've got the objects all right, but this is
not enough, because we have no way of picking them out. AxInf is what
allows this. So it's not (paradoxically) that this axiom guarantees
infinitely many numbers. It merely allows us to "lasso" objects that were
already there, and which would have been there, had the Axiom not been true.
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