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|Thus, by (IX), T is a consistent extension of the essentially undecidable theory T2, |
and therefore T is undecidable (cf. Theorem 3). Furthermore, every decision
procedure for T1 automatically yields a decision procedure for T. Hence T1 is ...
|Introduction A general method for establishing the undecidability of theories was |
developed in . Its basic ... There is a finitely axiomatizable and essentially
undecidable theory Q which is a fragment of the arithmetic of natural numbers.
|Now we can generalize our work on undecidable theories. Hereditarily and |
essentially undecidable theories A theory T is hereditarily undecidable if it is
undecidable and for every theory in the language of T, is undecidable. A theory T
|1.2, If a'theory T is compatible with a finitely axiomatizable and essentially |
undecidable theory 2, then T is undecidable. A still weaker fragment of number
theory, the theory R, was also introduced in . R is not finitely axiomatizable.
|by Alonzo Church in 1936; he showed that the system ofPrincipia Math- ematica |
is undecidable if it is w-consistent. ... The main new concept of the first part of the
book is that of a theory being essentially undecidable, which means that the ...
|We see thus that the equational theory of RA and all of its extensions mentioned |
in (xii)(/3) are examples of equational theories that are undecidable and dually
decidable; they could be referred to as theories which are essentially dually ...
|(This is the case for our axioms Q, and probably for any realistic set of arithmetic |
axioms.) Essential undecidability A theory is said to be essentially undecidable if
every consistent extension of it is undecidable. (By an extension of a theory T we
|Since an axiomatized theory resulting from an undecidable axiomatic theory |
through omission of a finite number of ... consistent extension of Peano's number
theory is undecidable, in short that this system is essentially undecidable, we
|It is easy to see that any complete axiomatizable theory is decidable. The |
classical result here is that any consistent theory has an extension to a complete
consistent theory. A theory is said to be essentially undecidable theory if it has no
|(c) Since T is not arithmetic it certainly is not r.e. So (T) is not a formal theory. (d) |
Let (T, ) be a ... ESSENTIAL UNrfECIDABILITY Tarski calls a theory (T) essentially
undecidable iff every consistent extension of (T) is undecidable. Consider now ...