Am I stupid for not understanding this exercise?
On page 440 of GEB, Hofstadter gives a puzzle to the reader, about two TNT proof-pairs. A pair of TNT Godel numbers forms a proof-pair when m is a valid TNT derivation and n is the last line of m.
Here are the two pairs. Hofstadter says that only one of them is valid and the other is not.
1.
m = 626,262,636,223,123,262,111,666,611,223,123,666,111,666
n = 123,666,111,666
2.
m = 626,262,636,223,123,262,111,666,611,223,333,262,636,123,262,111,666
n = 223,333,262,636,123,262,111,666
The last line of each m is the same as its n (611 is the Godel number symbolizing a line break) so that means that one of the two m numbers has to be an invalid derivation. Here is what they translate to:
1.
2.
But, BOTH of these derivations were used on pages 217 and 218 as examples of how to use TNT's rules of inference! And they're both derived in one step from the first line (which is an axiom of TNT) with no intermediary steps required! So which one of these derivations exactly is not valid? Or am I just retarded and missing something incredibly obvious?
On page 440 of GEB, Hofstadter gives a puzzle to the reader, about two TNT proof-pairs. A pair of TNT Godel numbers forms a proof-pair when m is a valid TNT derivation and n is the last line of m.
Here are the two pairs. Hofstadter says that only one of them is valid and the other is not.
1.
m = 626,262,636,223,123,262,111,666,611,223,123,666,111,666
n = 123,666,111,666
2.
m = 626,262,636,223,123,262,111,666,611,223,333,262,636,123,262,111,666
n = 223,333,262,636,123,262,111,666
The last line of each m is the same as its n (611 is the Godel number symbolizing a line break) so that means that one of the two m numbers has to be an invalid derivation. Here is what they translate to:
1.
2.
But, BOTH of these derivations were used on pages 217 and 218 as examples of how to use TNT's rules of inference! And they're both derived in one step from the first line (which is an axiom of TNT) with no intermediary steps required! So which one of these derivations exactly is not valid? Or am I just retarded and missing something incredibly obvious?
