Trivial Godel Numbering

A simple way to Godel number it seems is to change base.

So if we encode 0-9 as 0-9 and then up the base for each extra character. So for 6 extra characters we can use standard hexadecimal.

A = "="

B = "+"

C = "-"

D = "A"

E = "B"

F = ","

Problem that in the standard interpretation of algebra strings "00001" and "1" are the same number. For this reason keeping "0" meaning the same thing avoids issues with the detail.

Then the statement "1+1=2" corresponds to number 1B1A2 in hex which is the number 111010 in decimal.

Now we have two ways of creating Hex numbers. One way is the standard technique of switching bases so that 65535 in decimal becomes FFFF in Hex (call that the base interpretation B ). The other way is the direction substitution and therefore reinterpretation of arithmetic as Hex code (call that the arithmetic interpretation A). These two functions have inverses -A and -B. We can encode the use of the interpretations like this

"65535,B" produces FFFF (taking 65535 as a decimal number and base converting to hex)

while

"65535,-B" produces 415029 (taking 65535 as a hex number and base converting to decimal)

while

"1+1=2,A" produces 1B1A2 (taking 1+1=2 converting characters to character into arithmetic)

"1B1A2,-A" produces "1+1=2" (doing the opposite)

A,B,"," provide the characters of the metalevel and "-" is used at both levels to mean different things (in Hofstadter it is part of a tangled hierarchy)

so now the string "1+1=2,A" itself can be treated kind of meta-arithmetically and converted into a hex number using notation like 1+1=2,A,A (i.e. applying the arithmetic interpretation function to the text "1+1=2,A")

1+1=2,A,A produces 1B1A2FDFD

We can apply the inverse Base interpretation to that 1B1A2FDFD,-B to give the decimal number 7275216381

OK because of my choice of domains the decimal numbers can never map into the higher "meta" domains and so numbers always remain as numbers. Can redo.

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Now there are all sorts of amazing proof which I am only just beginning to understand one of which is the fixed point theorem. We know before we start that there is a fixed point statement which is encodes itself... it seems this is only true however if I include a variable in the encoding since otherwise the statement would have to encode its own number which is impossible. Now that is one version of the SRH - that a system doesn't have room for a fixed point without substitution... how does Godel get around this? 2Do

So a sum which encodes to its own value, or what is "a" and "b" in this expression:

"a+b" -> "a B b"

2Do

Very rich and intuitive investigation of meta levels using basic maths :-) Notice the problem of "expressive power" that meta levels always create more symbol complexity than their mesa levels (see below).

Looking for antonym for meta- seems I'm not the only one:

http://oxymoronredundancyparadoxtrap.blogspot.com/2007/12/meta-and-its-antonym.html

from the bottom up meta- comes from Greek meaning simply beyond. It seems the specific flavour of being beyond itself is recent (even partly due to Hofstadter!!). Hofstadter isn't a philosopher which is a shame because ecstasy means to "stand outside oneself" which illutrates the oxymoronic process of meta much better and especially challenges the self-contained notion of self that H et al find problematic. And "en-stasis" would therefore be the notion of self being self-contained (the usual ego illusion we all have - I am I etc). It is interesting that another rendering of "in" or middle is mesa (μέσα) which seems the perfect antonym for meta. Of course this meta-discussion of meta itself illustrates the meta nature of the word mesa. Nothing could ever call itself mesa (without being meta) so it is a kind of double oxymoronic word since it can only be used by another to refer to the nature of something that in itself isn't meta. This is the SRH at work big time.

Wiki here on "meta" (being a meta page itself it is oddly not noted in the page) http://en.wikipedia.org/wiki/Meta