Consistency, my dear Mr.
“There is no consistency.
Let us now briefly turn to norms of consistency.
We contribute a formal proof for its strong eventual consistency.
The first concerns the need for consistency proofs.
For Frege on the other hand, consistency can never guarantee existence.
This restricted ω-consistency is called 1-consistency.
Consistency, consistency, consistency!
These considerations suggest the following more general program for proving consistency:
According to Maderich, a concealer with a thick consistency is actually ideal for pigmentation.
In 1936 Gentzen was able to prove the consistency of arithmetic given certain nonfinitistic assumptions.
Thus if the consistency of L were deduced within L, this fact would not prove the consistency of L.
Although it has been demonstrated that behaviour is seldom entirely consistent, it also has been shown that it reflects considerable consistency.
A runnier consistency is great for slightly soaking into the cake and dripping down the sides, but you can also make a just-spoonable consistency to sit neatly on top of a cake.
One norm that does seem to be infringed by delusions is consistency, where this is intended both as consistency between the delusion and the person’s other beliefs, and consistency between the delusion and the person’s behavior.
But, for reasons explained above, such a consistency proof would have no mathematical value for Brouwer; and the best a classical mathematician can be said to be doing, according to the view Brouwer sketches, is to be giving relative consistency proofs.
Hilbert is clearly the winner in this debate, in the sense that roughly his conception of consistency is what one means today by “consistency” in the context of formal theories, and a near relative of his methodology for consistency-proofs is now standard.
What Gödel’s result showed was that there can be no absolute consistency proof of all of mathematics; hence work in proof theory after Gödel concentrated on relative results, both: relative to the system for which a consistency proof was given, and relative to the proof methods used.
In 1963, twenty-five years after Gödel’s proof of the consistency of the CH and the AC, relative to the consistency of ZF, Paul Cohen (1966) proved the consistency of the negation of the CH, and also of the negation of the AC, relative to the consistency of ZF.
Because for Hilbert the consistency of a set of sentences turns entirely on the overall structure they exhibit, while for Frege the consistency of the set of thoughts expressed turns additionally on the contents of the non-logical terms appearing in the sentences, on this account, Hilbert-consistency doesn’t imply Frege-consistency.
- the property of holding together and retaining its shape
- a harmonious uniformity or agreement among things or parts
- logical coherence and accordance with the facts
Example: a rambling argument that lacked any consistency
- (logic) an attribute of a logical system that is so constituted that none of the propositions deducible from the axioms contradict one another
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Because for Hilbert the consistency of a set of sentences turns entirely on the overall structure they exhibit while for Frege the consistency of the set of thoughts expressed turns additionally on the contents of the non-logical terms appearing in the sentences on this account Hilbert-consistency doesnt imply Frege-consistency