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Finite promise games and greedy clique sequences

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The finite promise games are a collection of mathematical games developed by American mathematician Harvey Friedman in 2009 which are used to develop a family of fast-growing functions , and . The greedy clique sequence is a graph theory concept, also developed by Friedman in 2010, which are used to develop fast-growing functions , and .

represents the theory of ZFC plus, the infinite family of axioms "there exists a strongly -Mahlo cardinal for all positive integers . and represents the theory of ZFC plus "for each , there is a strongly -Mahlo cardinal". represents the theory of ZFC plus, for each , "there is a -stationary Ramsey cardinal", and represents the theory of ZFC plus "for each , there is a strongly -stationary Ramsey cardinal". represents the theory of ZFC plus, for each , "there is a -huge cardinal", and represents the theory of ZFC plus "for each , there is a strongly -huge cardinal".

Finite promise games

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Each of the games is finite, predetermined in length, and has two players (Alice and Bob). At each turn, Alice chooses an integer or a number of integers (an offering) and the Bob has to make one of two kinds of promises restricting his future possible moves. In all games, Bob wins if and only if Bob has kept all of his promises.

Finite piecewise linear copy/invert games

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Here, is the set of integers, and is the set of non-negative integers. Here, all letters represent integers. We say that a map is piecewise linear if can be defined by various affine functions with integer coefficients on each of finitely many pieces, where each piece is defined by a finite set of linear inequalities with integer coefficients. For some piecewise linear map , a -inversion of is some such that . We then define the game for nonzero .

has rounds, and alternates between Alice and Bob. At every stage of the game, Alice is required to play , called her offering, which is either of the form or , where and are integers previously played by Bob. Bob is then required to either:

  • Accept , thereby playing and promising that there will be no -inversion of among the integers ever played by Bob. This promise applies to all past, present and future plays in the game.
  • Reject , thereby play a -inversion of and promising that is never played by Bob.

In RCA0, it can be proven that Bob always has a winning strategy for any given game. The game is a modified version where Bob is forced to accept all factorial offers by Alice . Bob always has a winning strategy for for sufficiently large , although this cannot be proven in any given consistent fragment of , and only . The function is the smallest such that Bob can win  for any  such that  and  are greater than or equal to  and all the following values are less than :

  • (the domain of is )
  • the number of pieces of
  • the absolute values of the coefficients of the inequalities in
  • the absolute values of the coefficients of the affine functions in

Finite polynomial copy/invert games

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Let be a polynomial with integer coefficients. A special -inversion at in consists of such that . We now define the game for nonzero , where are polynomials with integer coefficients. consists of alternating plays by Alice and Bob. At every stage of the game, Alice is required to play of the form , or , where is a -tuple of integers previously played by Bob. Bob is then required to either:

  • Accept , thereby playing and promising that there will be no special - or -inversion of among the integers ever played by Bob. This promise applies to all past, present and future plays in the game.
  • Reject , thereby playing a special - or -inversion of and promising that is never played by Bob.

Let be polynomials with integer coefficients. In RCA0, it can be proven that Bob always has a winning strategy for any given game. If are sufficiently large then Bob wins , which is where Bob is forced to accept all double factorials offered by Alice. However, once again, this cannot be proven in any given consistent fragment of , and only . The function is the smallest such that Bob can win  for any  such that  and  are greater than or equal to  and all the following values are less than :

  • (the domain of the polynomials is )
  • the degrees of and
  • the absolute values of the coefficients of and

Finite linear copy/invert games

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We say that are additively equivalent if and only if . For nonzero integers and , we define the game which consists of alternating rounds between Alice and Bob. At every stage of the game, Alice is required to play an integer of the form or , where are integers previously played by Bob. Bob is then required to either:

  • Accept , thereby playing and promising that cannot be written as , where is additively equivalent to some , and are integers played by Bob at various times.
  • Reject , thereby playing , where and is additively equivalent to some , and promises that is never played by Bob.

Let . In RCA0, it can be proven that Bob always has a winning strategy for any given game. Let . If is sufficiently large, then Bob wins , where Bob accepts all factorials offered by Alice. However, once again, this cannot be proven in any given consistent fragment of , and only . The function is the smallest such that Bob can win  for any  such that  is greater than or equal to , are positive and all the following values are less than :

  • the number of components of each vector in

Functions

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As shown by Friedman, the three functions ,  and  are extremely fast-growing, eventually dominating any functions provably recursive in any consistent fragment of (one of these is ZFC), but they are computable and provably total in .

Greedy clique sequences

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denotes the set of all tuples of rational numbers. We use subscripts to denote indexes into tuples (starting at 1) and angle brackets to denote concatenation of tuples, e.g. . Given , we define the upper shift of , denoted  to be the result of adding 1 to all its nonnegative components. Given , we say that  and are called order equivalent if and only if they have the same length and for all iff . A set  is order invariant iff for all order equivalent  and , .

Let  be a graph with vertices in . Let  be the set defined as follows: for every edge  in , their concatenation  is in . Then if  is order invariant, we say that  is order invariant. When  is order invariant,  has infinite edges. We are given , , and a simple graph (or a digraph in the case of upper shift greedy down clique sequences) with vertices in . We define a sequence  as a nonempty tuple  where . This is not a tuple but rather a tuple of tuples. When , is said to be an upper shift greedy clique sequence in  if it satisfies the following:

  • consists only of zeroes.
  • Let be an integer such that , or a positive integer if we allow infinite sequences. and let . Then, , is not an edge of , and .
  • is a clique in , i.e. contains as an edge every pair of vertices in .

When , is said to be an upper shift down greedy clique sequence in  if it satisfies the following:

  • consists only of zeroes.
  • Let be an integer such that , or a positive integer if we allow infinite sequences. and let . Then, or and is not an edge of ; and .
  • is a down clique in , i.e. for all and , is an edge of .

When , is said to be an extreme upper shift down greedy clique sequence in  if it satisfies the following:

  • consists only of zeroes.
  • Let be an integer such that , or a positive integer if we allow infinite sequences. and let . Then, or and is not an edge of ; and .
  • If , then .
  • If , then .
  • If and , then
  • is a down clique in

The thread of is a subsequence defined inductively like so:

Given a thread , we say that is open if . Using this Harvey Friedman defined three very powerful functions:

  • is the smallest such that every simple, order invariant graph  has an upper shift greedy clique sequence in  of length at most  with an open thread.
  • is the smallest such that every order invariant digraph  has an upper shift greedy down clique sequence in  of length at most  with an open thread.
  • is the smallest such that every order invariant digraph  has an extreme upper shift greedy down clique sequence in  of length at most  with an open thread.

and eventually dominate all functions provably recursive in , but are themselves provably recursive in . eventually dominates all functions provably recursive in , but is itself provably total in .

References

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  • Friedman, Harvey (2009-09-02). "Finite Promise Games". New York University. Retrieved 2021-10-02.
  • Friedman, Harvey (2009-12-30). "Upper Shift Greedy Clique Sequences and Large Cardinals 1". New York University. Retrieved 2021-10-02.
  • Friedman, Harvey (2010-01-01). "Terrifically and Extremely Long Finite Sequences". New York University. Retrieved 2021-10-02.