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I'll try to give an overview of what's going on. In the late 1800s Peano came up with his set of axioms for the integers. This set is second order which means
by yequalsx 16y ago
I'll try to give an overview of what's going on. In the late 1800s Peano came up with his set of axioms for the integers. This set is second order which means it isn't computable. Hilbert and others wanted a computable set of axioms so that one could, in theory, remove humans from the discovery of mathematical truth.
So the first order axioms were invented. Godel showed that the first order axioms are not strong enough to prove all true statements about the integers. There are infinitely distinct models of the first order integers. There is only one model of the second order integers.
In the proof of the Incompleteness Theorem Godel proves a statement about an integer which is not provable in the first order system.
Humans can work in the second order system but not computers. So, from my perspective there is something different about human thought patterns. I'm not an expert and would like to know what is wrong with my reasoning.
- Symmetry 16y agoOk, I'll confess that I'm not sure what you're talking about when you mention second order systems. However, Godel's incompleteness theorems (plural) were a lot more general than you're suggesting. First Theorem: "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory." Second Theorem: "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent." It wasn't that Godel proved that Peano arithmetic was incomplete that shocked mathematicians, it was that he showed that _no formal system_ could ever be complete as long as remained consistent. Of course maybe this second order system isn't consistent, in which case there's no problem. After all, I'm fairly convinced that human reason isn't always perfectly consistent, so it makes sense that we can avoid Godel's theorems. Of course, I see no reason we couldn't make a slightly inconsistent computer either.
- yequalsx 16y agoDefinitely his Incompleteness Theorem (I there are two of them and also a Completeness Theorem) was more general than what I mentioned but the shock was its application to arithmetic. The second order axioms are not recursively enumerable (think computable). It was the goal of Hilbert to find a computable system capable of proving (in theory) all true statements. There is no such computable system. One can find complete axiomatic systems of the integers. It's quite easy. Just take all true statements as the axiom system. The problem is that there is no computable method for determining in such a system whether or not a given statement is an axiom. Again, we humans can work in the second order axiom system but computers can't. So it appears there's a difference between human intelligence and computer intelligence.
- wlievens 16y agoI still don't understand what you mean by the "second order axiom system" that is somehow the domain only of humans...