SRH and Contradiction

Thinking this all through again on the train yesterday the way "outside" a formal system seems to be through contradiction.

A system functions "internally" and oblivious to anything else until a statement arises that contradicts an axiom (the starting data of the system). Now the system has a problem because if it can prove something one way and prove its contradiction another way then ... hmm what actually is the problem?

Here is a simple system of statements (the very assumption that they are to be taken together as 1 system is already a meta statement that cannot be encoded in the system) but in mesa terms (see previous post) where we (the meta-agent) allow the system to be self-contained (the mesa-agent):

system1{1.All horses have four legs 2.Spot is a horse 3.Spot has three legs.}

Actually when I get good enough at logic to do this stuff properly task 1 is to encode the limits of 1 system within another and explore that.

System1 has a problem because if 1&2 are true then 3 is false. If 1&3 are true 2 is false, and if 2&3 are true then 1 is false. One of these statements doesn't belong in the system therefore it belongs "outside" the system.

A contradiction forces the system open. This is Hegel because once the system is split then a meta level is opened up which must be greater than the system and so constitutes a new system, meta to the first system. This new system itself is then vulnerable to contradiction.

There are not always contradictory statements but once self-reference is possible there are always undecidable statements which achieve the same purpose.

This formal systems that seek consistency always need to expand - like this blog in fact. The only way to end this blog would be to accept inconsistencies... speculation but may be.