Re: Godel's proof, truth, reality, self-awareness, and all that jazz

On Sep 25, 10:42 am, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:

I've developed a set theory (Bit mapped set theory) which CAN NOT
serve as a foundation of mathematics, because every (hereditarily
finite) set in that theory is equivalent with a natural number. Thus
there is simply NO ROOM for e.g. ordered pairs in that theory -
supposing these could be distinguished from naturals.

Very strange comment. If A and B are hereditarily finite, then so is
A x B and hence you've *already* mapped A x B to a natural number.

I don't see that your theory is all that useful, but it does seem a
nice encoding of hereditarily finite sets as natural numbers. But, if
I may make a suggestion, your theory is very oddly expressed. Here's
what you say:

Formation rules:
0. {} is a set (the empty set)
1. if A is a set and B are sets then AB are sets
2. if A are sets then {A} is a set
3. There are no other ways to form (sets or) a set

Rule (1) is just strange: if A is a set and B are sets, then AB are
sets. This is a pretty idiosyncratic way of making (2) do what you
want. Why not drop (1) and write (2) as:

2'. if A1, A2, A3, ..., An are sets, then {A1,A2,A3,...,An} is a set.

Okay, so I had to use commas to make it readable, but there's no
particular reason *not* to use commas in this notation.

I have come to believe that this serves as an encoding for all
hereditarily finite sets, but I notice that you never prove this
claim. Why not?


See my previous comments on this system:

http://groups.google.com/group/sci.logic/browse_frm/thread/e02b54f8cd91e429/5c5960b236bafb83?lnk=gst&q=chas+%22are+sets%22&rnum=1#5c5960b236bafb83

It is /not/ a system of heriditary finite sets, it is a system
consisting of "is a set"s (/sets/ with a single element) and "are
sets"s (which are finite /sequences/ of two or more "is a set"s).

The main problem is that his system does not obey "A = B iff (x in A
iff x in B)" as it stands: the encoding for {{}, {{}}} is not the same
as the encoding for {{{}}, {}}, and equality is defined in terms of
equality of encoding. Thus we cannot make sense of the statement "A =
{x : P(x)}" in his system.

Cheers - Chas

.

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