By Miller G. A.

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Apart from this we shall have all the usual machinery of predicate calculus. And we could have lots of other things too. We could even have variables ranging over functions and so on; it will not make any difference at all. The proof will still go through. Indeed, when I get to the question of axioms, we shall see that we can throw in a few extra of these too. There will be certain restrictions here: if you throw in an infinite number of axioms, then you might not be able to prove Godel's theorem.

All these operations can be accomplished by synthesis from basic tasks such as 1-4, and it is preferable to give a diagram ofthe underlying structure of V rather than attempt to list hundreds of quadruples. , and markers are kept on the current state block in M and the scanned symbol block in P. ) The current state block will be followed by the 'current symbol block', 'current act block' and 'next state block'. *'I~ Move nshllo nexl quadruple t--- I("""'Iual If equal Read currenl aCI block If pnnl new symbol Replace scanned symbol block bY currenl act block If R I 'I'~ Move scanned symbol marker one block 10 righI, shifting ·10 rishl if -- Compare state block bepnninalhis quadruple wilh marked stale block If ··reached 41 Go 10 scanned symbol block, erue marker, and stop t Move marker on currenl state block 10 nexl slale block and go 10 leftmosl quadruple necessary Move scanned symbol marker one block 10 left.

Now I claim that ~ = (B, < is a model for I:, and that it is not isomorphic to Yo Why is ~ a model for I:? Each of the sentences of I: is true in ~ as you can see by inspection. None of these numbers is smaller than itself, so Vx ( l x < x) is true. No number is both smaller than and greater than another, so (ix) is true. Also < is transitive over B and any two elements of B are related, so (x) and (xi) are true in ~. 0 is a first element, so (xii) is true and each element has a successor and all but zero has an immediate predecessor, so (xiii) and (xiv) are true.

### Theorems Relating to the History of Mathematics by Miller G. A.

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