Thanks. I will look at it again tomorrow, but now I guess I have an approximate idea how it goes.
1) We can create various self-referential statements that syntaxtically seem completely innocent. They are composed from the same components and in the same way as ordinary (non-self-referential) statements. So, if the obviously innocent statements are allowed to be made, the clever schemes for creating self-referential statements will be allowed, too. -- This is the first thing to think about, separately from the rest.
2) Self-referential statements are allowed to speak about themselves, and also about me (a given axiomatic system). Thus they can abuse me in various ways which I can’t really avoid. Because even if I tried to refuse thinking about them, they can make themselves mean “Viliam refuses to think about me”, and provide me a step-by-step proof, which I would have to admit is correct. Then they make themselves something like “If Viliam says I am correct, then 2+2=3″, which I cannot avoid or deny, because both would make the statement true, so the only remaining option is to admit it, which means I have admitted that 2+2=3. -- This is another thing to think about., because I don’t feel really convinced by what I wrote. I realize that both avoiding to answer, or saying that the statement is incorrect would be wrong… and yet I somehow can’t imagine in detail how specifically could I say it was true.
Maybe a better intuition instead of seeing myself as an axiomatic system (which I am not) would be to imagine that at the beginning I precommit to use a specific PA-compatible list of rules… and then I follow the rules blindly, even when I see that the clever statements somehow force me to do things that are somehow wrong, but technically okay according to those rules.
Thanks. I will look at it again tomorrow, but now I guess I have an approximate idea how it goes.
1) We can create various self-referential statements that syntaxtically seem completely innocent. They are composed from the same components and in the same way as ordinary (non-self-referential) statements. So, if the obviously innocent statements are allowed to be made, the clever schemes for creating self-referential statements will be allowed, too. -- This is the first thing to think about, separately from the rest.
2) Self-referential statements are allowed to speak about themselves, and also about me (a given axiomatic system). Thus they can abuse me in various ways which I can’t really avoid. Because even if I tried to refuse thinking about them, they can make themselves mean “Viliam refuses to think about me”, and provide me a step-by-step proof, which I would have to admit is correct. Then they make themselves something like “If Viliam says I am correct, then 2+2=3″, which I cannot avoid or deny, because both would make the statement true, so the only remaining option is to admit it, which means I have admitted that 2+2=3. -- This is another thing to think about., because I don’t feel really convinced by what I wrote. I realize that both avoiding to answer, or saying that the statement is incorrect would be wrong… and yet I somehow can’t imagine in detail how specifically could I say it was true.
Maybe a better intuition instead of seeing myself as an axiomatic system (which I am not) would be to imagine that at the beginning I precommit to use a specific PA-compatible list of rules… and then I follow the rules blindly, even when I see that the clever statements somehow force me to do things that are somehow wrong, but technically okay according to those rules.