On 01 Aug 2011, at 01:40, Pzomby wrote:

The following quote is from the book “What is Mathematics Really?” by
Reuben Hersh

“0 (zero) is particularly nice.   It is the class of sets equivalent
to the set of all objects unequal to themselves!  No object is unequal
to itself, so 0 is the class of all empty sets.  But all empty sets
have the same members….none!  So they’re not merely equivalent to each
other…they are all the same set.  There’s only one empty set!  (A set
is characterized by its membership list.  There’s no way to tell one
empty membership list from another.  Therefore all empty sets are the
same thing!)

Once I have the empty sets, I can use a trick of Von Neumann as an
alternative way to construct the number 1.  Consider the class of all
empty sets.  This class has exactly one member: the unique empty set.
It’s a singleton.  ‘Out of nothing’ I have made a singleton set…a
“canonical representative” for the cardinal number 1.  1 is the class
of all singletons…all sets but with a single element.  To avoid
circularity: 1 is the class of all sets equivalent to the set whose
only element is the empty set.  Continuing, you get pairs, triplets,
and so on.  Von Neumann recursively constructs the whole set of
natural numbers out of sets of nothing.

….The idea of set…any collection of distinct objects…was so simple and
fundamental; it looked like a brick out of which all mathematics could
be constructed.  Even arithmetic could be downgraded (or upgraded)
from primary to secondary rank, for the natural numbers could be
constructed, as we have just seen, from nothing…ie., the empty set…by
operations of set theory.”

Any comments or opinions on whether this theory is the basis for the
natural numbers and their relations as is described in the quote

To use set theory for studying the numbers is like taking an airbus 380 to go to the grocery. Set theory is too big, and it flatten the concepts (unlike categories which sharpen them, when used carefully).

Now, ZF, the Zermelo-Fraenkel formal set theory, is a cute example of (arithmetical) little Löbian Universal Machine, and is handy as an example of a very imaginative machine capable of handling most of PA's theology.

PA is much weaker than ZF, but like the guy in the chinese room which can simulate a chinese talking person, PA can simulate (emulate, even) ZF. Well, even RA can do that.

But set theories and most toposes give too much larger ontology, when you assume comp. They do have epistemological roles, to be sure, and they do prove *much* more arithmetical truth than PA. But then many other theories do.

There is no real problem if you prefer to adopt set theoretical realism, instead of arithmetical realism, when assuming comp. This will not change anything in the extraction of theology and physics from comp, except you will meet even more people criticizing your ontology (as being too much big!).

If you like set, you can take the theory of hereditary finite sets, which can be shown equivalent with PA.

Well, to be sure, putting infinite sets in the ontology can inadvertently leads to treachery in the explanation of why machines (finite beings) can believe in infinite sets.



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