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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tfr1onlemsucaccv 6301* | Lemma for tfr1on 6310. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlembacc 6302* | Lemma for tfr1on 6310. Each element of is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembxssdm 6303* | Lemma for tfr1on 6310. The union of is defined on all elements of . (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembfn 6304* | Lemma for tfr1on 6310. The union of is a function defined on . (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlembex 6305* | Lemma for tfr1on 6310. The set exists. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlemubacc 6306* | Lemma for tfr1on 6310. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemex 6307* | Lemma for tfr1on 6310. (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemaccex 6308* |
We can define an acceptable function on any element of .
As with many of the transfinite recursion theorems, we have hypotheses that state that is a function and that it is defined up to . (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemres 6309* | Lemma for tfr1on 6310. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.) |
recs | ||
Theorem | tfr1on 6310* | Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfri1dALT 6311* |
Alternate proof of tfri1d 6295 in terms of tfr1on 6310.
Although this does show that the tfr1on 6310 proof is general enough to also prove tfri1d 6295, the tfri1d 6295 proof is simpler in places because it does not need to deal with being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.) |
recs | ||
Theorem | tfrcllemssrecs 6312* | Lemma for tfrcl 6324. The union of functions acceptable for tfrcl 6324 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucfn 6313* | We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6324. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucaccv 6314* | Lemma for tfrcl 6324. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllembacc 6315* | Lemma for tfrcl 6324. Each element of is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembxssdm 6316* | Lemma for tfrcl 6324. The union of is defined on all elements of . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembfn 6317* | Lemma for tfrcl 6324. The union of is a function defined on . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembex 6318* | Lemma for tfrcl 6324. The set exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemubacc 6319* | Lemma for tfrcl 6324. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemex 6320* | Lemma for tfrcl 6324. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemaccex 6321* |
We can define an acceptable function on any element of .
As with many of the transfinite recursion theorems, we have hypotheses that state that is a function and that it is defined up to . (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemres 6322* | Lemma for tfr1on 6310. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.) |
recs | ||
Theorem | tfrcldm 6323* | Recursion is defined on an ordinal if the characteristic function satisfies a closure hypothesis up to a suitable point. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcl 6324* | Closure for transfinite recursion. As with tfr1on 6310, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfri1 6325* |
Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of
[TakeutiZaring] p. 47, with an
additional condition.
The condition is that is defined "everywhere", which is stated here as . Alternately, would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfri2 6326* | Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 6325). Here we show that the function has the property that for any function satisfying that condition, the "next" value of is recursively applied to all "previous" values of . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfri3 6327* | Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule ( as described at tfri1 6325). Finally, we show that is unique. We do this by showing that any class with the same properties of that we showed in parts 1 and 2 is identical to . (Contributed by Jim Kingdon, 4-May-2019.) |
recs | ||
Theorem | tfrex 6328* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 6329 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 6330* |
Define a recursive definition generator on (the class of ordinal
numbers) 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 operation (especially when df-recs 6265
that it is built on is also eliminated). But once we get past this
hurdle, definitions that would otherwise be recursive become relatively
simple. In classical logic it would be easier to divide this definition
into cases based on whether the domain of is zero, a successor, or
a limit ordinal. Cases do not (in general) work that way in
intuitionistic logic, so instead we choose a definition which takes the
union of all the results of the characteristic function for ordinals in
the domain of .
This means that this definition has the expected
properties for increasing and continuous ordinal functions, which
include ordinal addition and multiplication.
For finite recursion we also define df-frec 6351 and for suitable characteristic functions df-frec 6351 yields the same result as restricted to , as seen at frecrdg 6368. Note: We introduce 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 Jim Kingdon, 19-May-2019.) |
recs | ||
Theorem | rdgeq1 6331 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 6332 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 6333 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 6334* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 6335* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 6336* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 6337 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 6338 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 6339 | 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 6346; in cases like df-oadd 6380 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 6340* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 6341* | Value of the recursive definition generator. Lemma for rdgival 6342 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 6342* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 6343 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 6344* |
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 6345. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 6345* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6424 and omsuc 6432. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 6346* | 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.) (Revised by Jim Kingdon, 13-Apr-2022.) |
Theorem | rdg0 6347 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 6348 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 6349 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 6350 | Extend class notation with the finite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 6351* |
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 6265
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 6357 and frecsuc 6367.
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 4576. 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 6368, this definition and df-irdg 6330 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 6352 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 6353 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 6354 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | frecfun 6355 | Finite recursion produces a function. See also frecfnom 6361 which also states that the domain of that function is but which puts conditions on and . (Contributed by Jim Kingdon, 13-Feb-2022.) |
frec | ||
Theorem | nffrec 6356 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 6357 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 6358* | The class abstraction from df-frec 6351 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frecabcl 6359* | The class abstraction from df-frec 6351 exists. Unlike frecabex 6358 the function only needs to be defined on , not all sets. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 21-Mar-2022.) |
Theorem | frectfr 6360* |
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 6295 or tfri2d 6296,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 6361* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | freccllem 6362* | Lemma for freccl 6363. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.) |
recs frec | ||
Theorem | freccl 6363* | Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.) |
frec | ||
Theorem | frecfcllem 6364* | Lemma for frecfcl 6365. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.) |
recs frec | ||
Theorem | frecfcl 6365* | Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.) |
frec | ||
Theorem | frecsuclem 6366* | Lemma for frecsuc 6367. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.) |
frec frec | ||
Theorem | frecsuc 6367* | The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.) |
frec frec | ||
Theorem | frecrdg 6368* |
Transfinite recursion restricted to omega.
Given a suitable characteristic function, df-frec 6351 produces the same results as df-irdg 6330 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
frec | ||
Syntax | c1o 6369 | Extend the definition of a class to include the ordinal number 1. |
Syntax | c2o 6370 | Extend the definition of a class to include the ordinal number 2. |
Syntax | c3o 6371 | Extend the definition of a class to include the ordinal number 3. |
Syntax | c4o 6372 | Extend the definition of a class to include the ordinal number 4. |
Syntax | coa 6373 | Extend the definition of a class to include the ordinal addition operation. |
Syntax | comu 6374 | Extend the definition of a class to include the ordinal multiplication operation. |
Syntax | coei 6375 | Extend the definition of a class to include the ordinal exponentiation operation. |
↑_{o} | ||
Definition | df-1o 6376 | Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) |
Definition | df-2o 6377 | Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) |
Definition | df-3o 6378 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-4o 6379 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-oadd 6380* | Define the ordinal addition operation. (Contributed by NM, 3-May-1995.) |
Definition | df-omul 6381* | Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.) |
Definition | df-oexpi 6382* |
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 6383 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
Theorem | 1oex 6384 | Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.) |
Theorem | 2on 6385 | Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | 2on0 6386 | Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) |
Theorem | 3on 6387 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | 4on 6388 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | df1o2 6389 | Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.) |
Theorem | df2o3 6390 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | df2o2 6391 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
Theorem | 1n0 6392 | Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
Theorem | xp01disj 6393 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
Theorem | xp01disjl 6394 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) |
Theorem | ordgt0ge1 6395 | Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
Theorem | ordge1n0im 6396 | An ordinal greater than or equal to 1 is nonzero. (Contributed by Jim Kingdon, 26-Jun-2019.) |
Theorem | el1o 6397 | Membership in ordinal one. (Contributed by NM, 5-Jan-2005.) |
Theorem | dif1o 6398 | Two ways to say that is a nonzero number of the set . (Contributed by Mario Carneiro, 21-May-2015.) |
Theorem | 2oconcl 6399 | Closure of the pair swapping function on . (Contributed by Mario Carneiro, 27-Sep-2015.) |
Theorem | 0lt1o 6400 | Ordinal zero is less than ordinal one. (Contributed by NM, 5-Jan-2005.) |
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