Thoughts you cannot think

[nortexoid]

Lucas and Penrose have argued that Godel's (1st incompleteness) theorem refutes mechanism.

All it shows that for any recursively axiomatizable formal system with Peano arithmetic, we can construct a sentence that is neither provable nor disprovable, but is nonetheless true. Call that sentence G (for Godel). But this is only under the assumption that such a formal system is consistent. If it is not consistent, then every formula is a theorem of the system, and so too is G. The problem is that we cannot prove in the system that such a system is consistent - proved by Godel's 2nd incompleteness theorem. So all we have is that

CONs-->G

which means, if S (some formal arithmetical system) is consistent, then G is true. But we cannot prove CONs, so we cannot infer G. So how do we know that G is true? Because our minds which are not recursively axiomatizable (like formal systems and machines) can see that's true by some other unaxiomatizable means. So while S cannot prove G and so G is not in S (and is thus not a theorem of S, and so isn't true in S), we know it's true. Therefore, Godel's theorem doesn't show that there are things that humans cannot think of, it shows that there are formulas that machines (or formal systems) are incapable of knowing, or proving.

There is a big difference between any formal arithmetical system and "true" arithmetic, for which all arithmetical formulas, including G, are true. "True" arithmetic is complete (but not recursively axiomatizable), while any given formal arithmetical system isn't (proved by G).

 

[jhu]

if there are thoughts that we cannot think, how would we know them?

 

[Smilin]

A bit off topic and possibly more info than you want to know about me..

I'm using nicotine patches which have this really odd side effect of bizare and realistic dreams that completely fill the night. It's sometimes pleasant but sometimes not. Your brain just reacts really oddly to having nicotine while you sleep. Not something you can usually manage with cigarrettes.

Anyway so I'm having this apocalyptic dream that people all over the earth are being struck by some terrifying thought that leaves them screaming and unable to stop. Something incomprehensible but once they have it, their minds are gone. Bizare I tell ya.

 

[Smilin]

Originally posted by: nortexoid
Lucas and Penrose have argued that Godel's (1st incompleteness) theorem refutes mechanism.

All it shows that for any recursively axiomatizable formal system with Peano arithmetic, we can construct a sentence that is neither provable nor disprovable, but is nonetheless true. Call that sentence G (for Godel). But this is only under the assumption that such a formal system is consistent. If it is not consistent, then every formula is a theorem of the system, and so too is G. The problem is that we cannot prove in the system that such a system is consistent - proved by Godel's 2nd incompleteness theorem. So all we have is that

CONs-->G

which means, if S (some formal arithmetical system) is consistent, then G is true. But we cannot prove CONs, so we cannot infer G. So how do we know that G is true? Because our minds which are not recursively axiomatizable (like formal systems and machines) can see that's true by some other unaxiomatizable means. So while S cannot prove G and so G is not in S (and is thus not a theorem of S, and so isn't true in S), we know it's true. Therefore, Godel's theorem doesn't show that there are things that humans cannot think of, it shows that there are formulas that machines (or formal systems) are incapable of knowing, or proving.

There is a big difference between any formal arithmetical system and "true" arithmetic, for which all arithmetical formulas, including G, are true. "True" arithmetic is complete (but not recursively axiomatizable), while any given formal arithmetical system isn't (proved by G).

I didn't follow that, and something grey and spongy came out of my ear the second time I read through it.

I'm going to think about this some more once I get it back in.

 

[CSMR]

Interesting stuff nortexoid

 

[jhu]

Originally posted by: Smilin
A bit off topic and possibly more info than you want to know about me..

I'm using nicotine patches which have this really odd side effect of bizare and realistic dreams that completely fill the night. It's sometimes pleasant but sometimes not. Your brain just reacts really oddly to having nicotine while you sleep. Not something you can usually manage with cigarrettes.

Anyway so I'm having this apocalyptic dream that people all over the earth are being struck by some terrifying thought that leaves them screaming and unable to stop. Something incomprehensible but once they have it, their minds are gone. Bizare I tell ya.

have you watched the movie "pi"?

 

[effee]

wow this is deep.

 

[Smilin]

Originally posted by: jhu

have you watched the movie "pi"?

No, what's it about?

Edit:

Ah. I'll check it out.
http://en.wikipedia.org/wiki/Pi_(movie)

Sounds kinda interesting. Reminds me of some kinda deep abstract thoughts I've had about fractals+heaven+hell before.

 

[Armitage]

Something I've always found interesting is the studies of brain function based on individuals with brain damage. They basically take somebody that has had a certain part of their brain dmaged in some way, and see what impact that has on them to determine what different parts ofthe brain do. What really struck me is how our perception of the world is shaped by the structure of our brain. For example, I think there was one person who basically couldn't "perceive" horizontal lines. They could read script letters fine, but had trouble with block letters. So, are there other aspects of our environment which our brains simply aren't wired right to perceive?

 

[silverpig]

Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

What if you can prove that your system can't prove its own consistency? Is that therefore a proof that it is consistent?

 

[BrentUnitedMem]

In response to: "if there are thoughts that we cannot think, how would we know them"

I have an example of this...

It is impossible to think about the largest prime number, however you can proove quite simply that the number of elements in the set of prime numbers is infinite.

 

[nortexoid]

Originally posted by: silverpig

Stated more colloquially, any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.

What if you can prove that your system can't prove its own consistency? Is that therefore a proof that it is consistent?

that is exactly Gödel's 2nd incompleteness theorem - that for any sufficiently strong system S, if S is consistent, it cannot be proved in S that S is consistent.