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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | smo0 6101 | The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smofvon 6102 | If is a strictly monotone ordinal function, and is in the domain of , then the value of the function at is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.) |
Theorem | smoel 6103 | If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiun 6104* | The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Theorem | smoiso 6105 | If is an isomorphism from an ordinal onto , which is a subset of the ordinals, then is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.) |
Theorem | smoel2 6106 | A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.) |
Syntax | crecs 6107 | Notation for a function defined by strong transfinite recursion. |
recs | ||
Definition | df-recs 6108* |
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 6173 for more details on why
this definition is desirable. Unlike df-irdg 6173 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 6138 and
tfri2d 6139 for the primary contract of this definition.
(Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | recseq 6109 | Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs recs | ||
Theorem | nfrecs 6110 | Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
recs | ||
Theorem | tfrlem1 6111* | 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 6112* | Lemma for transfinite recursion. This lemma just changes some bound variables in for later use. (Contributed by Jim Kingdon, 5-Jul-2019.) |
Theorem | tfrlem3a 6113* | 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 6114* | 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 6115* | Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlem4 6116* | 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 6117* | 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 6118* | Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.) |
recs | ||
Theorem | tfrlem6 6119* | 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 6120* | 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 6121* | 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 6122* | 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 6123 | Transfinite recursion produces a function. (Contributed by Jim Kingdon, 20-Aug-2021.) |
recs | ||
Theorem | tfr2a 6124 | A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
recs | ||
Theorem | tfr0dm 6125 | Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.) |
recs | ||
Theorem | tfr0 6126 | Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
recs | ||
Theorem | tfrlemisucfn 6127* | We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Theorem | tfrlemisucaccv 6128* | We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibacc 6129* | Each element of is an acceptable function. Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 14-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibxssdm 6130* | The union of is defined on all ordinals. Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibfn 6131* | The union of is a function defined on . Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemibex 6132* | The set exists. Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiubacc 6133* | The union of satisfies the recursion rule (lemma for tfrlemi1 6135). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemiex 6134* | Lemma for tfrlemi1 6135. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.) |
Theorem | tfrlemi1 6135* |
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 6136* | The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.) |
recs | ||
Theorem | tfrexlem 6137* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Theorem | tfri1d 6138* |
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 6139* | 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 6168). 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 6140* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3ag 6112 but for tfr1on 6153 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.) |
Theorem | tfr1onlem3 6141* | Lemma for transfinite recursion. This lemma changes some bound variables in (version of tfrlem3 6114 but for tfr1on 6153 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.) |
Theorem | tfr1onlemssrecs 6142* | Lemma for tfr1on 6153. The union of functions acceptable for tfr1on 6153 is a subset of recs. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucfn 6143* | We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6153. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlemsucaccv 6144* | Lemma for tfr1on 6153. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.) |
recs | ||
Theorem | tfr1onlembacc 6145* | Lemma for tfr1on 6153. Each element of is an acceptable function. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembxssdm 6146* | Lemma for tfr1on 6153. The union of is defined on all elements of . (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlembfn 6147* | Lemma for tfr1on 6153. The union of is a function defined on . (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlembex 6148* | Lemma for tfr1on 6153. The set exists. (Contributed by Jim Kingdon, 14-Mar-2022.) |
recs | ||
Theorem | tfr1onlemubacc 6149* | Lemma for tfr1on 6153. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 15-Mar-2022.) |
recs | ||
Theorem | tfr1onlemex 6150* | Lemma for tfr1on 6153. (Contributed by Jim Kingdon, 16-Mar-2022.) |
recs | ||
Theorem | tfr1onlemaccex 6151* |
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 6152* | Lemma for tfr1on 6153. 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 6153* | 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 6154* |
Alternate proof of tfri1d 6138 in terms of tfr1on 6153.
Although this does show that the tfr1on 6153 proof is general enough to also prove tfri1d 6138, the tfri1d 6138 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 6155* | Lemma for tfrcl 6167. The union of functions acceptable for tfrcl 6167 is a subset of recs. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucfn 6156* | We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6167. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllemsucaccv 6157* | Lemma for tfrcl 6167. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 24-Mar-2022.) |
recs | ||
Theorem | tfrcllembacc 6158* | Lemma for tfrcl 6167. Each element of is an acceptable function. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembxssdm 6159* | Lemma for tfrcl 6167. The union of is defined on all elements of . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembfn 6160* | Lemma for tfrcl 6167. The union of is a function defined on . (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllembex 6161* | Lemma for tfrcl 6167. The set exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemubacc 6162* | Lemma for tfrcl 6167. The union of satisfies the recursion rule. (Contributed by Jim Kingdon, 25-Mar-2022.) |
recs | ||
Theorem | tfrcllemex 6163* | Lemma for tfrcl 6167. (Contributed by Jim Kingdon, 26-Mar-2022.) |
recs | ||
Theorem | tfrcllemaccex 6164* |
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 6165* | Lemma for tfr1on 6153. 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 6166* | 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 6167* | Closure for transfinite recursion. As with tfr1on 6153, 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 6168* |
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 6169* | 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 6168). 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 6170* | 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 6168). 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 6171* | The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
recs | ||
Syntax | crdg 6172 | Extend class notation with the recursive definition generator, with characteristic function and initial value . |
Definition | df-irdg 6173* |
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 6108
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 6194 and for suitable characteristic functions df-frec 6194 yields the same result as restricted to , as seen at frecrdg 6211. 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 6174 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgeq2 6175 | Equality theorem for the recursive definition generator. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.) |
Theorem | rdgfun 6176 | The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Theorem | rdgtfr 6177* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgruledefgg 6178* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgruledefg 6179* | The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexggg 6180 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgexgg 6181 | The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Theorem | rdgifnon 6182 | 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 6189; in cases like df-oadd 6223 either presumably could work). (Contributed by Jim Kingdon, 13-Jul-2019.) |
Theorem | rdgifnon2 6183* | The recursive definition generator is a function on ordinal numbers. (Contributed by Jim Kingdon, 14-May-2020.) |
Theorem | rdgivallem 6184* | Value of the recursive definition generator. Lemma for rdgival 6185 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.) |
Theorem | rdgival 6185* | Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Theorem | rdgss 6186 | Subset and recursive definition generator. (Contributed by Jim Kingdon, 15-Jul-2019.) |
Theorem | rdgisuc1 6187* |
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 6188. (Contributed by Jim Kingdon, 9-Jun-2019.) |
Theorem | rdgisucinc 6188* |
Value of the recursive definition generator at a successor.
This can be thought of as a generalization of oasuc 6265 and omsuc 6273. (Contributed by Jim Kingdon, 29-Aug-2019.) |
Theorem | rdgon 6189* | 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 6190 | The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Theorem | rdg0g 6191 | The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Theorem | rdgexg 6192 | The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Syntax | cfrec 6193 | Extend class notation with the finite recursive definition generator, with characteristic function and initial value . |
frec | ||
Definition | df-frec 6194* |
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 6108
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 6200 and frecsuc 6210.
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 4447. 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 6211, this definition and df-irdg 6173 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 6195 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | freceq2 6196 | Equality theorem for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec frec | ||
Theorem | frecex 6197 | Finite recursion produces a set. (Contributed by Jim Kingdon, 20-Aug-2021.) |
frec | ||
Theorem | frecfun 6198 | Finite recursion produces a function. See also frecfnom 6204 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 6199 | Bound-variable hypothesis builder for the finite recursive definition generator. (Contributed by Jim Kingdon, 30-May-2020.) |
frec | ||
Theorem | frec0g 6200 | The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.) |
frec |
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