Discrete exterior calculus: Difference between revisions
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[[David Hilbert|Hilbert]]'s '''epsilon calculus''' is an extension of a [[formal language]] by the epsilon operator, where the epsilon operator substitutes for [[quantification|quantifiers]] in that language as a method leading to a [[consistency proof | proof of consistency]] for the extended formal language. The ''epsilon operator'' and ''epsilon substitution method'' are typically applied to a [[first-order logic|first-order predicate calculus]], followed by a showing of consistency. The epsilon-extended calculus is further extended and generalized to cover those mathematical objects, classes, and categories for which there is a desire to show consistency, building on previously-shown consistency at earlier levels.<ref>Stanford, overview paragraphs</ref> | |||
==Epsilon operator== | |||
===Hilbert notation=== | |||
For any formal language ''L'', extend ''L'' by adding the epsilon operator to redefine quantification: | |||
*<math> (\exists x) A(x)\ \equiv \ A(\epsilon x\ A) </math> | |||
*<math> (\forall x) A(x)\ \equiv \ A(\epsilon x\ (\neg A)) </math> | |||
The intended interpretation of ε''x'' ''A'' is ''some x'' that satisfies ''A'', if it exists. In other words, ε''x'' ''A'' returns some term ''t'' such that ''A''(''t'') is true, otherwise it returns some default or arbitrary term. If more than one term can satisfy ''A'', then any one of these terms (which make ''A'' true) can be [[Axiom of Choice|chosen]], non-deterministically. Equality is required to be defined under ''L'', and the only rules required for ''L'' extended by the epsilon operator are modus ponens and the substitution of ''A''(''t'') to replace ''A''(''x'') for any term ''t''.<ref>Stanford, the epsilon calculus paragraphs</ref> | |||
===Bourbaki notation=== | |||
In [[tau-square]] notation from [[Bourbaki|N. Bourbaki's]] ''Theory of Sets'', the quantifiers are defined as follows: | |||
*<math> (\exists x) A(x)\ \equiv \ (\tau_x(A)|x)A </math> | |||
*<math> (\forall x) A(x)\ \equiv \ \neg (\tau_x(\neg A)|x)\neg A\ \equiv \ (\tau_x(\neg A)|x)A</math> | |||
where ''A'' is a relation in ''L'', ''x'' is a variable, and <math>\tau_x(A)</math> juxtaposes a <math>\tau</math> at the front of ''A'', replaces all instances of ''x'' with <math>\square</math>, and links them back to <math>\tau</math>. Then let ''Y'' be an assembly, ''(Y|x)A'' denotes the replacement of all variables ''x'' in ''A'' with ''Y''. | |||
This notation is equivalent to the Hilbert notation and is read the same. It is used by Bourbaki to define [[cardinal assignment]] since he does not use the [[axiom of replacement]]. | |||
==Modern approaches== | |||
[[David Hilbert#Hilbert's program |Hilbert's Program]] for mathematics was to justify those [[formal system]]s as consistent in relation to constructive or semi-constructive systems. While Gödel's results on incompleteness mooted Hilbert's Program to a great extent, modern researchers find the epsilon calculus to provide alternatives for approaching proofs of systemic consistency as described in the epsilon substitution method. | |||
===Epsilon substitution method=== | |||
A theory to be checked for consistency is first embedded in an appropriate epsilon calculus. Second, a process is developed for re-writing quantified theorems to be expressed in terms of epsilon operations via the epsilon substitution method. Finally, the process must be shown to normalize the re-writing process, so that the re-written theorems satisfy the axioms of the theory.<ref>Stanford, more recent developments paragraphs</ref> | |||
==References== | |||
* {{IEP|ep-calc|Epsilon Calculi}} | |||
*{{cite book | last = Moser | first = G. | coauthors = R. Zach | title = The Epsilon Calculus (Tutorial) | location = Berlin | publisher = Springer-Verlag | oclc = 108629234}} | |||
*Stanford Encyclopedia of Philosophy (online). [http://plato.stanford.edu/entries/epsilon-calculus/ ''The Epsilon Calculus''] | |||
*{{cite book | last = Bourbaki | first = N. | title = Theory of Sets | location = Berlin | publisher = Springer-Verlag | isbn = 3-540-22525-0}} | |||
==Notes== | |||
<references/> | |||
[[Category:Systems of formal logic]] | |||
[[Category:Mathematical logic]] | |||
[[Category:Proof theory|*]] | |||
Revision as of 01:25, 23 July 2013
Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. The epsilon operator and epsilon substitution method are typically applied to a first-order predicate calculus, followed by a showing of consistency. The epsilon-extended calculus is further extended and generalized to cover those mathematical objects, classes, and categories for which there is a desire to show consistency, building on previously-shown consistency at earlier levels.[1]
Epsilon operator
Hilbert notation
For any formal language L, extend L by adding the epsilon operator to redefine quantification:
The intended interpretation of εx A is some x that satisfies A, if it exists. In other words, εx A returns some term t such that A(t) is true, otherwise it returns some default or arbitrary term. If more than one term can satisfy A, then any one of these terms (which make A true) can be chosen, non-deterministically. Equality is required to be defined under L, and the only rules required for L extended by the epsilon operator are modus ponens and the substitution of A(t) to replace A(x) for any term t.[2]
Bourbaki notation
In tau-square notation from N. Bourbaki's Theory of Sets, the quantifiers are defined as follows:
where A is a relation in L, x is a variable, and juxtaposes a at the front of A, replaces all instances of x with , and links them back to . Then let Y be an assembly, (Y|x)A denotes the replacement of all variables x in A with Y.
This notation is equivalent to the Hilbert notation and is read the same. It is used by Bourbaki to define cardinal assignment since he does not use the axiom of replacement.
Modern approaches
Hilbert's Program for mathematics was to justify those formal systems as consistent in relation to constructive or semi-constructive systems. While Gödel's results on incompleteness mooted Hilbert's Program to a great extent, modern researchers find the epsilon calculus to provide alternatives for approaching proofs of systemic consistency as described in the epsilon substitution method.
Epsilon substitution method
A theory to be checked for consistency is first embedded in an appropriate epsilon calculus. Second, a process is developed for re-writing quantified theorems to be expressed in terms of epsilon operations via the epsilon substitution method. Finally, the process must be shown to normalize the re-writing process, so that the re-written theorems satisfy the axioms of the theory.[3]
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- Stanford Encyclopedia of Philosophy (online). The Epsilon Calculus
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534