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In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or functions of the theory. [1]
The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof.
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations.
The proof-theoretic ordinal of such a theory is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals for which there exists a notation in Kleene's sense such that proves that is an ordinal notation. Equivalently, it is the supremum of all ordinals such that there exists a recursive relation on (the set of natural numbers) that well-orders it with ordinal and such that proves transfinite induction of arithmetical statements for .
Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z2 to "prove well-ordered", we instead construct an ordinal notation with order type . can now work with various transfinite induction principles along , which substitute for reasoning about set-theoretic ordinals.
However, some pathological notation systems exist that are unexpectedly difficult to work with. For example, Rathjen gives a primitive recursive notation system that is well-founded iff PA is consistent, [2]p. 3 despite having order type - including such a notation in the ordinal analysis of PA would result in the false equality .
For any theory that's both -axiomatizable and -sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by -soundness. Thus the proof-theoretic ordinal of a -sound theory that has a axiomatization will always be a (countable) recursive ordinal, that is, less than the Church–Kleene ordinal . [2]Theorem 2.21
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations.
What is the proof-theoretic ordinal of full second-order arithmetic? [4]
Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes , full second-order arithmetic () and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF.
Ordinal | First-order arithmetic | Second-order arithmetic | Kripke-Platek set theory | Type theory | Constructive set theory | Explicit mathematics | |
---|---|---|---|---|---|---|---|
, | |||||||
, | |||||||
, | , | ||||||
[1] | , | ||||||
, [7]p. 13 | [7]p. 13, [7]p. 13 | ||||||
[8] [7]p. 13 | [9]: 40 | ||||||
[7]p. 13 | [7]p. 13, , [7]p. 13, [10]p. 8 | [11]p. 869 | |||||
, [12] [13]: 8 | |||||||
[14]p. 959 | |||||||
, [15] [13] , [16]: 7 [15]p. 17, [15]p. 5 | |||||||
, [15]p. 52 | |||||||
, [17] | |||||||
, [18]p. 17, [18]p. 17 | [19]p. 140, [19]p. 140, [19]p. 140, [10]p. 8 | [11]p. 870 | |||||
[10]p. 27, [10]p. 27 | |||||||
[20]p.9 | |||||||
[2] | |||||||
, [21] , [18]p. 22, [18]p. 22, [22] | , , , [23] [24]p. 26 | [11]p. 878, [11]p. 878 | , | ||||
[25]p.13 | |||||||
[26] | |||||||
[16]: 7 | |||||||
[16]: 7 | |||||||
, [27] | [28]p.1167, [28]p.1167 | ||||||
[27] | [28]p.1167, [28]p.1167 | ||||||
[27]: 11 | |||||||
[29]p.233, [29]p.233 | [30]p.276 | [30]p.276 | |||||
[29]p.233, [16] | [30]p.277 | [30]p.277 | |||||
[16]: 7 | |||||||
, [31] [16]: 7 | |||||||
[16]: 7 | |||||||
[3] | [10]p. 8 | , [2] , [11]p. 869 | |||||
[10]p. 31, [10]p. 31, [10]p. 31 | |||||||
[32] | |||||||
[10]p. 33, [10]p. 33, [10]p. 33 | |||||||
[4] | , [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26 | [24]p. 26, [24]p. 26 | |||||
[4]p. 28 | [4]p. 28, | ||||||
[33] | |||||||
[34]p. 14 | |||||||
[35] | |||||||
[33] | |||||||
[33] | |||||||
[5] | |||||||
[4]p. 28 | |||||||
[4]p. 28, | |||||||
[6] | |||||||
, , [36] | , | ||||||
, , ,, , [36]: 72 | , [36]: 72 [36]: 72 | , [36]: 72 | |||||
, , [36]: 72 | [36]: 72 | ||||||
, , [36]: 72 | [36]: 72 | ||||||
, , [36]: 72 | |||||||
, , [36]: 72 | , [36]: 72 | ||||||
, , [36]: 72 | , [36]: 72 | ||||||
[7] | [4]p. 28, | ||||||
[37]: 38 | |||||||
[8] | |||||||
[9] | |||||||
[10] | |||||||
[11] | [38] | [38] | |||||
[12] | [39] | ||||||
[13] | [40] | ||||||
[14] | [40] | ||||||
[41] | , [41] [42] | ||||||
[41] | , | ||||||
[43] | , | ||||||
? | [43] | , | [44] |
This is a list of symbols used in this table:
This is a list of the abbreviations used in this table:
A superscript zero indicates that -induction is removed (making the theory significantly weaker).
{{
citation}}
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{{
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: CS1 maint: bot: original URL status unknown (
link)This article includes a list of general
references, but it lacks sufficient corresponding
inline citations. (September 2021) |
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or functions of the theory. [1]
The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof.
Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient portion of arithmetic to make statements about ordinal notations.
The proof-theoretic ordinal of such a theory is the supremum of the order types of all ordinal notations (necessarily recursive, see next section) that the theory can prove are well founded—the supremum of all ordinals for which there exists a notation in Kleene's sense such that proves that is an ordinal notation. Equivalently, it is the supremum of all ordinals such that there exists a recursive relation on (the set of natural numbers) that well-orders it with ordinal and such that proves transfinite induction of arithmetical statements for .
Some theories, such as subsystems of second-order arithmetic, have no conceptualization or way to make arguments about transfinite ordinals. For example, to formalize what it means for a subsystem of Z2 to "prove well-ordered", we instead construct an ordinal notation with order type . can now work with various transfinite induction principles along , which substitute for reasoning about set-theoretic ordinals.
However, some pathological notation systems exist that are unexpectedly difficult to work with. For example, Rathjen gives a primitive recursive notation system that is well-founded iff PA is consistent, [2]p. 3 despite having order type - including such a notation in the ordinal analysis of PA would result in the false equality .
For any theory that's both -axiomatizable and -sound, the existence of a recursive ordering that the theory fails to prove is well-ordered follows from the bounding theorem, and said provably well-founded ordinal notations are in fact well-founded by -soundness. Thus the proof-theoretic ordinal of a -sound theory that has a axiomatization will always be a (countable) recursive ordinal, that is, less than the Church–Kleene ordinal . [2]Theorem 2.21
Friedman's grand conjecture suggests that much "ordinary" mathematics can be proved in weak systems having this as their proof-theoretic ordinal.
This ordinal is sometimes considered to be the upper limit for "predicative" theories.
The Kripke-Platek or CZF set theories are weak set theories without axioms for the full powerset given as set of all subsets. Instead, they tend to either have axioms of restricted separation and formation of new sets, or they grant existence of certain function spaces (exponentiation) instead of carving them out from bigger relations.
What is the proof-theoretic ordinal of full second-order arithmetic? [4]
Most theories capable of describing the power set of the natural numbers have proof-theoretic ordinals that are so large that no explicit combinatorial description has yet been given. This includes , full second-order arithmetic () and set theories with powersets including ZF and ZFC. The strength of intuitionistic ZF (IZF) equals that of ZF.
Ordinal | First-order arithmetic | Second-order arithmetic | Kripke-Platek set theory | Type theory | Constructive set theory | Explicit mathematics | |
---|---|---|---|---|---|---|---|
, | |||||||
, | |||||||
, | , | ||||||
[1] | , | ||||||
, [7]p. 13 | [7]p. 13, [7]p. 13 | ||||||
[8] [7]p. 13 | [9]: 40 | ||||||
[7]p. 13 | [7]p. 13, , [7]p. 13, [10]p. 8 | [11]p. 869 | |||||
, [12] [13]: 8 | |||||||
[14]p. 959 | |||||||
, [15] [13] , [16]: 7 [15]p. 17, [15]p. 5 | |||||||
, [15]p. 52 | |||||||
, [17] | |||||||
, [18]p. 17, [18]p. 17 | [19]p. 140, [19]p. 140, [19]p. 140, [10]p. 8 | [11]p. 870 | |||||
[10]p. 27, [10]p. 27 | |||||||
[20]p.9 | |||||||
[2] | |||||||
, [21] , [18]p. 22, [18]p. 22, [22] | , , , [23] [24]p. 26 | [11]p. 878, [11]p. 878 | , | ||||
[25]p.13 | |||||||
[26] | |||||||
[16]: 7 | |||||||
[16]: 7 | |||||||
, [27] | [28]p.1167, [28]p.1167 | ||||||
[27] | [28]p.1167, [28]p.1167 | ||||||
[27]: 11 | |||||||
[29]p.233, [29]p.233 | [30]p.276 | [30]p.276 | |||||
[29]p.233, [16] | [30]p.277 | [30]p.277 | |||||
[16]: 7 | |||||||
, [31] [16]: 7 | |||||||
[16]: 7 | |||||||
[3] | [10]p. 8 | , [2] , [11]p. 869 | |||||
[10]p. 31, [10]p. 31, [10]p. 31 | |||||||
[32] | |||||||
[10]p. 33, [10]p. 33, [10]p. 33 | |||||||
[4] | , [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26, [24]p. 26 | [24]p. 26, [24]p. 26 | |||||
[4]p. 28 | [4]p. 28, | ||||||
[33] | |||||||
[34]p. 14 | |||||||
[35] | |||||||
[33] | |||||||
[33] | |||||||
[5] | |||||||
[4]p. 28 | |||||||
[4]p. 28, | |||||||
[6] | |||||||
, , [36] | , | ||||||
, , ,, , [36]: 72 | , [36]: 72 [36]: 72 | , [36]: 72 | |||||
, , [36]: 72 | [36]: 72 | ||||||
, , [36]: 72 | [36]: 72 | ||||||
, , [36]: 72 | |||||||
, , [36]: 72 | , [36]: 72 | ||||||
, , [36]: 72 | , [36]: 72 | ||||||
[7] | [4]p. 28, | ||||||
[37]: 38 | |||||||
[8] | |||||||
[9] | |||||||
[10] | |||||||
[11] | [38] | [38] | |||||
[12] | [39] | ||||||
[13] | [40] | ||||||
[14] | [40] | ||||||
[41] | , [41] [42] | ||||||
[41] | , | ||||||
[43] | , | ||||||
? | [43] | , | [44] |
This is a list of symbols used in this table:
This is a list of the abbreviations used in this table:
A superscript zero indicates that -induction is removed (making the theory significantly weaker).
{{
citation}}
: CS1 maint: bot: original URL status unknown (
link)
{{
citation}}
: CS1 maint: bot: original URL status unknown (
link)