Re: Axiom of union

On Sep 19, 5:04 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
hagman wrote:

(One has to think about the Axiom of Infinity for a moment,
as it allows the set in question to have some additional elements,
but everything that you can *prove* an element of the INF set
has that property of being "hereditary empty").

Allow me to add to this what I'm dreaming about. The hereditary sets are
built up with extensionality, pairing & union from the empty set alone.

There is a bit of ambiguity in your wording here. I might, for
example, /not/ consider that x exists, y exists, thereore I can
"create" the set {x,y}.

Strictly speaking, the axioms don't say anything about "creating" or
"building"; they simply assert that the fact that x exists and y
exists /logically implies/ that {x,y} /also/ exists, rather than
"causes {x,y} to come into existence, when it didn't exist before".

If we could prove now: that these sets map uniquely on any map of bits,
up and down, in a digital computer of arbitrary size.

One can of course provide /a/ bijective mapping (the heriditary sets
are countable; so we can take the bits to collectively represent a
natural and extend the mapping from the bijection).

But there is no /unique/ such mapping, just as there is no unique
mapping of the bits in a computer to the naturals (simply apply any
permutation of the naturals with finite support to the ordering of the
bits).

Then we have IMHO
proven that the hereditary sets are necessary...

why "necessary"?

... as well as sufficient for
representing the world of computation, as seen through the eyes of e.g.
David Petry. (As far as the static part of it - memory without time -
is considered.)


I suppose it all depends on what you mean by "the world of
computation" and "represent".

Certainly the /objects/ of computer /calculations/ can be "represented
by", i.e., bijected with, /any/ countable set; for example, the
naturals, or the set of all finite strings over the alphabet {"H",
"L"}, or the collection of heridtarily finite sets, or for that matter
any countable set S of infinite ordinals e.g., {w, w+1, ...}, and so
on.

But in the case of heriditary sets for example, /computing/ the
representation corresponding to "f(x)", given the representations for /
specific/ "f" and "x", is distinct from simply /choosing/ a
representation for "f", "x", and the value to be associated with
"f(x)" in that case.

It is not enough to simply assert "x is represented by 1, f is
represented by 2; /therefore/ f(x) is represented by 17". We still
must /prove/ that the computation of (or calculation of or algorithm
for, etc.) "f(x)" is /consistent/ with the /interpretation/ of these
representations /as/ heriditary sets.

The actual computation in question in practice boils down to some
sequence of operations in boolean algebra; because under the /current/
designs the various gates etc. represent elements of some large but
finite field of order 2^n, aka, GF(2^n). We then exhibit a
mathematical /isomorphism/ between these operations and the evaluation
of "f(x)" that is /provably consistent/ with our /interpretation/ of f
and x /as/ heriditary sets.

Where I find David's argument becomes weak is that he approves of only
certain /interpretations/ of what those elements of GF(2^n) are,
themselves, /allowed/ to further represent when we want to consider
them as something /other/ than elements of GF(2^n).

Sure, interpreting them as representing the naturals is ok, and as you
suggest here finite heriditary sets (I think he'd concur).

But it seems strange to me to suggest that there is a "permissible"
interpretation where we assign 0 to "the set {} having no elements"
and 1 to "the set {{}}", such that we can produce a calculation
resulting in 17 that allows us to correctly infer that "{} is a member
of {{}}"; but we are /not/ allowed to assign 0 to "the set of even
naturals E" and 1 to "{E}", and perform an identical calculation
resulting in 17 that allows us to correctly infer that "E is a member
of {E}", solely because E (being infinite) is not part of the "world
of computation".

The question of which /interpretations/ of the elements of GF(2^n) are
permissible and therefore part of DP's "world of computation", and
which are not, is not primarily a mathematical one (although one of
course must provide some isomorphism to claim that one has an
interpretation); instead it's a philosophical one (he rejects the
axiom of infinity). I think his philosophical argument is not really
very coherent, because he hasn't really addressed these issues of
interpretation head-on.

Cheers - Chas

.

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