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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rdgtfr 5901* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 5902* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 5903* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 5904 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 5905 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 5906 | The recursive definition generator is a function on ordinal numbers. The condition states that the characteristic function is defined for all sets (being defined for all ordinals might be enough if being used in a manner similar to rdgon 5913; in cases like df-oadd 5944 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 5907* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 5908* | Value of the recursive definition generator. Lemma for rdgival 5909 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 5909* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 5910 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 5911* |
One way of describing the value of the recursive definition generator at
a successor. There is no condition on the characteristic function
other than
. Given that, the resulting expression
encompasses both the expected successor term
but also terms that correspond to
the initial value and to limit ordinals
.
If we add conditions on the characteristic function, we can show tighter results such as rdgisucinc 5912. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 5912* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 5983 and omsuc 5990. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 5913* | Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdg0 5914 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 5915 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 5916 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 5917 | Extend class notation with the fnite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 5918* |
Define a recursive definition generator on (the class of finite
ordinals) with characteristic function and initial value .
This rather amazing operation allows us to define, with compact direct
definitions, functions that are usually defined in textbooks only with
indirect self-referencing recursive definitions. A recursive definition
requires advanced metalogic to justify - in particular, eliminating a
recursive definition is very difficult and often not even shown in
textbooks. On the other hand, the elimination of a direct definition is
a matter of simple mechanical substitution. The price paid is the
daunting complexity of our frec operation (especially when df-recs 5861
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple; see frec0g 5922 and frecsuc 5930.
Unlike with transfinite recursion, finite recurson can readily divide definitions and proofs into zero and successor cases, because even without excluded middle we have theorems such as nn0suc 4270. The analogous situation with transfinite recursion - being able to say that an ordinal is zero, successor, or limit - is enabled by excluded middle and thus is not available to us. For the characteristic functions which satisfy the conditions given at frecrdg 5931, this definition and df-irdg 5897 restricted to produce the same result. Note: We introduce frec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style. (Contributed by Mario Carneiro and Jim Kingdon, 10-Aug-2019.) |
frec recs | ||
Theorem | freceq1 5919 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 5920 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | nffrec 5921 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 5922 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 5923* | The class abstraction from df-frec 5918 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frectfr 5924* |
Lemma to connect transfinite recursion theorems with finite recursion.
That is, given the conditions
and on
frec ,
we want to be able to apply tfri1d 5890 or tfri2d 5891,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 5925* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | frecsuclem1 5926* | Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 13-Aug-2019.) |
frec recs | ||
Theorem | frecsuclemdm 5927* | Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs | ||
Theorem | frecsuclem2 5928* | Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.) |
recs frec | ||
Theorem | frecsuclem3 5929* | Lemma for frecsuc 5930. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecsuc 5930* | The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.) |
frec frec | ||
Theorem | frecrdg 5931* |
Transfinite recursion restricted to omega.
Given a suitable characteristic function, df-frec 5918 produces the same results as df-irdg 5897 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
frec | ||
Theorem | freccl 5932* | Closure for finite recursion. (Contributed by Jim Kingdon, 25-May-2020.) |
frec | ||
Syntax | c1o 5933 | Extend the definition of a class to include the ordinal number 1. |
Syntax | c2o 5934 | Extend the definition of a class to include the ordinal number 2. |
Syntax | c3o 5935 | Extend the definition of a class to include the ordinal number 3. |
Syntax | c4o 5936 | Extend the definition of a class to include the ordinal number 4. |
Syntax | coa 5937 | Extend the definition of a class to include the ordinal addition operation. |
Syntax | comu 5938 | Extend the definition of a class to include the ordinal multiplication operation. |
Syntax | coei 5939 | Extend the definition of a class to include the ordinal exponentiation operation. |
↑_{𝑜} | ||
Definition | df-1o 5940 | Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) |
Definition | df-2o 5941 | Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) |
Definition | df-3o 5942 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-4o 5943 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-oadd 5944* | Define the ordinal addition operation. (Contributed by NM, 3-May-1995.) |
Definition | df-omul 5945* | Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.) |
Definition | df-oexpi 5946* |
Define the ordinal exponentiation operation.
This definition is similar to a conventional definition of exponentiation except that it defines ↑_{𝑜} to be for all , in order to avoid having different cases for whether the base is or not. (Contributed by Mario Carneiro, 4-Jul-2019.) |
↑_{𝑜} | ||
Theorem | 1on 5947 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
Theorem | 2on 5948 | Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | 2on0 5949 | Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) |
Theorem | 3on 5950 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | 4on 5951 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | df1o2 5952 | Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.) |
Theorem | df2o3 5953 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | df2o2 5954 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
Theorem | 1n0 5955 | Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
Theorem | xp01disj 5956 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
Theorem | ordgt0ge1 5957 | Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
Theorem | ordge1n0im 5958 | An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.) |
Theorem | el1o 5959 | Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) |
Theorem | dif1o 5960 | Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.) |
Theorem | 2oconcl 5961 | Closure of the pair swapping function on . (Contributed by Mario Carneiro, 27-Sep-2015.) |
Theorem | 0lt1o 5962 | Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.) |
Theorem | oafnex 5963 | The characteristic function for ordinal addition is defined everywhere. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Theorem | sucinc 5964* | Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) |
Theorem | sucinc2 5965* | Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.) |
Theorem | fnoa 5966 | Functionality and domain of ordinal addition. (Contributed by NM, 26-Aug-1995.) (Proof shortened by Mario Carneiro, 3-Jul-2019.) |
Theorem | oaexg 5967 | Ordinal addition is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | omfnex 5968* | The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Theorem | fnom 5969 | Functionality and domain of ordinal multiplication. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 3-Jul-2019.) |
Theorem | omexg 5970 | Ordinal multiplication is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Theorem | fnoei 5971 | Functionality and domain of ordinal exponentiation. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oeiexg 5972 | Ordinal exponentiation is a set. (Contributed by Mario Carneiro, 3-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oav 5973* | Value of ordinal addition. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | omv 5974* | Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Theorem | oeiv 5975* | Value of ordinal exponentiation. (Contributed by Jim Kingdon, 9-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oa0 5976 | Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | om0 5977 | Ordinal multiplication with zero. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | oei0 5978 | Ordinal exponentiation with zero exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
↑_{𝑜} | ||
Theorem | oacl 5979 | Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.) |
Theorem | omcl 5980 | Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.) |
Theorem | oeicl 5981 | Closure law for ordinal exponentiation. (Contributed by Jim Kingdon, 26-Jul-2019.) |
↑_{𝑜} | ||
Theorem | oav2 5982* | Value of ordinal addition. (Contributed by Mario Carneiro and Jim Kingdon, 12-Aug-2019.) |
Theorem | oasuc 5983 | Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | omv2 5984* | Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Theorem | onasuc 5985 | Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | oa1suc 5986 | Addition with 1 is same as successor. Proposition 4.34(a) of [Mendelson] p. 266. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Theorem | o1p1e2 5987 | 1 + 1 = 2 for ordinal numbers. (Contributed by NM, 18-Feb-2004.) |
Theorem | oawordi 5988 | Weak ordering property of ordinal addition. (Contributed by Jim Kingdon, 27-Jul-2019.) |
Theorem | oaword1 5989 | An ordinal is less than or equal to its sum with another. Part of Exercise 5 of [TakeutiZaring] p. 62. (Contributed by NM, 6-Dec-2004.) |
Theorem | omsuc 5990 | Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Theorem | onmsuc 5991 | Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nna0 5992 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
Theorem | nnm0 5993 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
Theorem | nnasuc 5994 | Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nnmsuc 5995 | Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nna0r 5996 | Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | nnm0r 5997 | Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | nnacl 5998 | Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nnmcl 5999 | Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Theorem | nnacli 6000 | is closed under addition. Inference form of nnacl 5998. (Contributed by Scott Fenton, 20-Apr-2012.) |
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