Godel's Proof

[daly]

Godel showed that it is impossible to formally prove the consistency of a system of arithmetic WITHIN that system of arithmetic.

Given a static system that is clearly true.

However, I claim a self-modifying system can extend itself to "look down" on the initial system of arithmetic.

I'm not sure how to show this yet. My intuition is leaning toward using the standard trick of towers of types. Is it possible to show a system of arithmetic is consistent using higher order types?

Which raises the question of self-modifying Turing machines. This leads to the concept of a meta-Turing machine.

Not only must the "instructions of the machine" be "on the tape" so they can be modified but the "structure of the hardware" also has to be "on the tape" so that changing the "structure" changes the way the machine operates. A simple implementation can be created in an FPGA which generates modifications to its own Verilog definition and reloads itself, becoming a fundamentally different machine.

[timonoko]

That is "The Most Beautiful Program Ever Written" - issue. Because I am stupid, I have to think very hard. Are we defining a language using the language itself or what?

https://youtu.be/OyfBQmvr2Hc

[suture]

The first order theory of arithmetic is provably consistent within ZFC. ZFC can’t prove it’s consistency but a “higher” system containing ZFC can prove that ZFC is consistent. So on and so forth.