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Type | Label | Description |
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Statement | ||
Theorem | tfri2d 6201* | 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 6230). 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 | tfr1onlem3ag 6202* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3ag 6174 but for tfr1on 6215 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Theorem | tfr1onlem3 6203* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3 6176 but for tfr1on 6215 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.) |
Theorem | tfr1onlemssrecs 6204* | Lemma for tfr1on 6215. The union of functions acceptable for tfr1on 6215 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucfn 6205* | We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6215. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucaccv 6206* | Lemma for tfr1on 6215. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlembacc 6207* | Lemma for tfr1on 6215. Each element of is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembxssdm 6208* | Lemma for tfr1on 6215. The union of is defined on all elements of . (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembfn 6209* | Lemma for tfr1on 6215. The union of is a function defined on . (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlembex 6210* | Lemma for tfr1on 6215. The set exists. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlemubacc 6211* | Lemma for tfr1on 6215. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemex 6212* | Lemma for tfr1on 6215. (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemaccex 6213* |
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 6214* | Lemma for tfr1on 6215. 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 6215* | 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 6216* |
Alternate proof of tfri1d 6200 in terms of tfr1on 6215.
Although this does show that the tfr1on 6215 proof is general enough to also prove tfri1d 6200, the tfri1d 6200 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 6217* | Lemma for tfrcl 6229. The union of functions acceptable for tfrcl 6229 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucfn 6218* | We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6229. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucaccv 6219* | Lemma for tfrcl 6229. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllembacc 6220* | Lemma for tfrcl 6229. Each element of is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembxssdm 6221* | Lemma for tfrcl 6229. The union of is defined on all elements of . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembfn 6222* | Lemma for tfrcl 6229. The union of is a function defined on . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembex 6223* | Lemma for tfrcl 6229. The set exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemubacc 6224* | Lemma for tfrcl 6229. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemex 6225* | Lemma for tfrcl 6229. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemaccex 6226* |
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 6227* | Lemma for tfr1on 6215. 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 6228* | 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 6229* | Closure for transfinite recursion. As with tfr1on 6215, 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 6230* |
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 6231* | 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 6230). 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 6232* | 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 6230). 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 6233* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 6234 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 6235* |
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 6170
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 6256 and for suitable characteristic functions df-frec 6256 yields the same result as restricted to , as seen at frecrdg 6273. 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 6236 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 6237 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 6238 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 6239* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 6240* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 6241* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 6242 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 6243 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 6244 | 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 6251; in cases like df-oadd 6285 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 6245* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 6246* | Value of the recursive definition generator. Lemma for rdgival 6247 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 6247* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 6248 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 6249* |
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 6250. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 6250* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6328 and omsuc 6336. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 6251* | 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 6252 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 6253 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 6254 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 6255 | Extend class notation with the finite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 6256* |
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 6170
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 6262 and frecsuc 6272.
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 4488. 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 6273, this definition and df-irdg 6235 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 6257 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 6258 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 6259 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | frecfun 6260 | Finite recursion produces a function. See also frecfnom 6266 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 6261 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 6262 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 6263* | The class abstraction from df-frec 6256 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frecabcl 6264* | The class abstraction from df-frec 6256 exists. Unlike frecabex 6263 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 6265* |
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 6200 or tfri2d 6201,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 6266* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | freccllem 6267* | Lemma for freccl 6268. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.) |
recs frec | ||
Theorem | freccl 6268* | Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.) |
frec | ||
Theorem | frecfcllem 6269* | Lemma for frecfcl 6270. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.) |
recs frec | ||
Theorem | frecfcl 6270* | Finite recursion yields a function on the natural numbers. (Contributed by Jim Kingdon, 30-Mar-2022.) |
frec | ||
Theorem | frecsuclem 6271* | Lemma for frecsuc 6272. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 29-Mar-2022.) |
frec frec | ||
Theorem | frecsuc 6272* | The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.) |
frec frec | ||
Theorem | frecrdg 6273* |
Transfinite recursion restricted to omega.
Given a suitable characteristic function, df-frec 6256 produces the same results as df-irdg 6235 restricted to . Presumably the theorem would also hold if were changed to . (Contributed by Jim Kingdon, 29-Aug-2019.) |
frec | ||
Syntax | c1o 6274 | Extend the definition of a class to include the ordinal number 1. |
Syntax | c2o 6275 | Extend the definition of a class to include the ordinal number 2. |
Syntax | c3o 6276 | Extend the definition of a class to include the ordinal number 3. |
Syntax | c4o 6277 | Extend the definition of a class to include the ordinal number 4. |
Syntax | coa 6278 | Extend the definition of a class to include the ordinal addition operation. |
Syntax | comu 6279 | Extend the definition of a class to include the ordinal multiplication operation. |
Syntax | coei 6280 | Extend the definition of a class to include the ordinal exponentiation operation. |
↑o | ||
Definition | df-1o 6281 | Define the ordinal number 1. (Contributed by NM, 29-Oct-1995.) |
Definition | df-2o 6282 | Define the ordinal number 2. (Contributed by NM, 18-Feb-2004.) |
Definition | df-3o 6283 | Define the ordinal number 3. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-4o 6284 | Define the ordinal number 4. (Contributed by Mario Carneiro, 14-Jul-2013.) |
Definition | df-oadd 6285* | Define the ordinal addition operation. (Contributed by NM, 3-May-1995.) |
Definition | df-omul 6286* | Define the ordinal multiplication operation. (Contributed by NM, 26-Aug-1995.) |
Definition | df-oexpi 6287* |
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 6288 | Ordinal 1 is an ordinal number. (Contributed by NM, 29-Oct-1995.) |
Theorem | 1oex 6289 | Ordinal 1 is a set. (Contributed by BJ, 4-Jul-2022.) |
Theorem | 2on 6290 | Ordinal 2 is an ordinal number. (Contributed by NM, 18-Feb-2004.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Theorem | 2on0 6291 | Ordinal two is not zero. (Contributed by Scott Fenton, 17-Jun-2011.) |
Theorem | 3on 6292 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | 4on 6293 | Ordinal 3 is an ordinal number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Theorem | df1o2 6294 | Expanded value of the ordinal number 1. (Contributed by NM, 4-Nov-2002.) |
Theorem | df2o3 6295 | Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Theorem | df2o2 6296 | Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.) |
Theorem | 1n0 6297 | Ordinal one is not equal to ordinal zero. (Contributed by NM, 26-Dec-2004.) |
Theorem | xp01disj 6298 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by NM, 2-Jun-2007.) |
Theorem | xp01disjl 6299 | Cartesian products with the singletons of ordinals 0 and 1 are disjoint. (Contributed by Jim Kingdon, 11-Jul-2023.) |
Theorem | ordgt0ge1 6300 | Two ways to express that an ordinal class is positive. (Contributed by NM, 21-Dec-2004.) |
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