Re: the return of the master : tommy1729

Transfer Principle <lwalke3@xxxxxxxxx> writes:

A few months ago, I was also concerned as to how to do the
Powerset in mereology. When I asked tommy1729 about this,
his reply was that higher cardinality are formed by using
sets of functions. In particular, the set of all functions
from set X to set Y is ordinarily written Y^X, and this is
the usual definition of cardinal exponentiation. And so we
can have card(Y^X)>card(X).

So in lieu of a Powerset Axiom, we can something like:

Axiom:
For every set X and set Y, the set Y^X exists.

What is a function in TST?

In set theory, a function is a particular set and hence an object of
the theory. We can thus discuss the set of functions X -> Y.

I don't see how a function is an object of (the ill-defined) TST, so
this axiom appears meaningless.

Notice that both this and tommy1729's Fixed Point Schema
go back to the concept of _functions_ (set of _functions_
from X to Y, fixed point of a _function_). In earlier
posts, tommy1729 has stated that both ordered pairs and
functions exist in his theory. I have to go back and dig
up the axioms which assert that all ordered pairs and
functions exist, since they are critical to TST.

Yeah, you could add an ad hoc axiom to the effect that ordered pairs
exist[1]. But Tommy's already given us a list of seven axioms (two of
which are meaningless) and called it TST. Unless he says that TST
includes distinct axioms regarding ordered pairs, I suppose that it
doesn't.

Of course, we haven't seen the definition of cardinality yet either,
but this also depends on having functions as objects. More
importantly, we need to know how many functions exist from X -> Y.
Once we've proved that ordered pairs exist, it does not follow that
X x Y exists nor that every subset of X x Y exists. For the latter
you need a powerset. Oops!

Footnotes:
[1] I vaguely recall you suggesting something to this effect quite
some time ago.

--
Jesse F. Hughes
"I get to make things move just by saying a few things. When I post
now the math world has to tremble, even if it does so quietly, hoping
that no one else notices." -- James S. Harris has the power.
.

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