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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | crecs 6201 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 6202* |
Define a function recs on , the class of ordinal
numbers, by transfinite recursion given a rule which sets the next
value given all values so far. See df-irdg 6267 for more details on why
this definition is desirable. Unlike df-irdg 6267 which restricts the
update rule to use only the previous value, this version allows the
update rule to use all previous values, which is why it is
described
as "strong", although it is actually more primitive. See tfri1d 6232 and
tfri2d 6233 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 6203 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 6204 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 6205* | A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Contributed by NM, 23-Mar-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlem3ag 6206* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by Jim Kingdon, 5-Jul-2019.) |
Theorem | tfrlem3a 6207* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3 6208* | Lemma for transfinite recursion. Let be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in for later use. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem3-2d 6209* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 6210* | Lemma for transfinite recursion. is the class of all "acceptable" functions, and is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.) |
Theorem | tfrlem5 6211* | Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.) |
Theorem | recsfval 6212* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 6213* | Lemma for transfinite recursion. The union of all acceptable functions is a relation. (Contributed by NM, 8-Aug-1994.) (Revised by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem7 6214* | Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.) |
recs | ||
Theorem | tfrlem8 6215* | Lemma for transfinite recursion. The domain of recs is ordinal. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Alan Sare, 11-Mar-2008.) |
recs | ||
Theorem | tfrlem9 6216* | Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.) |
recs recs recs | ||
Theorem | tfrfun 6217 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
recs | ||
Theorem | tfr2a 6218 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0dm 6219 | Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.) |
recs | ||
Theorem | tfr0 6220 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 6221* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 6222* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 6223* | Each element of is an acceptable function. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 6224* | The union of is defined on all ordinals. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 6225* | The union of is a function defined on . Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 6226* | The set exists. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 6227* | The union of satisfies the recursion rule (lemma for tfrlemi1 6229). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 6228* | Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 6229* |
We can define an acceptable function on any ordinal.
As with many of the transfinite recursion theorems, we have a hypothesis that states that is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi14d 6230* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 6231* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 6232* |
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 | tfri2d 6233* | 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 6262). 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 6234* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3ag 6206 but for tfr1on 6247 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Theorem | tfr1onlem3 6235* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3 6208 but for tfr1on 6247 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.) |
Theorem | tfr1onlemssrecs 6236* | Lemma for tfr1on 6247. The union of functions acceptable for tfr1on 6247 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucfn 6237* | We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6247. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucaccv 6238* | Lemma for tfr1on 6247. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlembacc 6239* | Lemma for tfr1on 6247. Each element of is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembxssdm 6240* | Lemma for tfr1on 6247. The union of is defined on all elements of . (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembfn 6241* | Lemma for tfr1on 6247. The union of is a function defined on . (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlembex 6242* | Lemma for tfr1on 6247. The set exists. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlemubacc 6243* | Lemma for tfr1on 6247. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemex 6244* | Lemma for tfr1on 6247. (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemaccex 6245* |
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 6246* | Lemma for tfr1on 6247. 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 6247* | 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 6248* |
Alternate proof of tfri1d 6232 in terms of tfr1on 6247.
Although this does show that the tfr1on 6247 proof is general enough to also prove tfri1d 6232, the tfri1d 6232 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 6249* | Lemma for tfrcl 6261. The union of functions acceptable for tfrcl 6261 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucfn 6250* | We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6261. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucaccv 6251* | Lemma for tfrcl 6261. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllembacc 6252* | Lemma for tfrcl 6261. Each element of is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembxssdm 6253* | Lemma for tfrcl 6261. The union of is defined on all elements of . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembfn 6254* | Lemma for tfrcl 6261. The union of is a function defined on . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembex 6255* | Lemma for tfrcl 6261. The set exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemubacc 6256* | Lemma for tfrcl 6261. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemex 6257* | Lemma for tfrcl 6261. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemaccex 6258* |
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 6259* | Lemma for tfr1on 6247. 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 6260* | 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 6261* | Closure for transfinite recursion. As with tfr1on 6247, 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 6262* |
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 6263* | 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 6262). 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 6264* | 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 6262). 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 6265* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 6266 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 6267* |
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 6202
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 6288 and for suitable characteristic functions df-frec 6288 yields the same result as restricted to , as seen at frecrdg 6305. 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 6268 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 6269 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 6270 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 6271* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 6272* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 6273* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 6274 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 6275 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 6276 | 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 6283; in cases like df-oadd 6317 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 6277* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 6278* | Value of the recursive definition generator. Lemma for rdgival 6279 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 6279* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 6280 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 6281* |
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 6282. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 6282* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6360 and omsuc 6368. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 6283* | 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 6284 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 6285 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 6286 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 6287 | Extend class notation with the finite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 6288* |
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 6202
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 6294 and frecsuc 6304.
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 4518. 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 6305, this definition and df-irdg 6267 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 6289 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 6290 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 6291 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | frecfun 6292 | Finite recursion produces a function. See also frecfnom 6298 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 6293 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 6294 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec | ||
Theorem | frecabex 6295* | The class abstraction from df-frec 6288 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.) |
Theorem | frecabcl 6296* | The class abstraction from df-frec 6288 exists. Unlike frecabex 6295 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 6297* |
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 6232 or tfri2d 6233,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Theorem | frecfnom 6298* | The function generated by finite recursive definition generation is a function on omega. (Contributed by Jim Kingdon, 13-May-2020.) |
frec | ||
Theorem | freccllem 6299* | Lemma for freccl 6300. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.) |
recs frec | ||
Theorem | freccl 6300* | Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.) |
frec |
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