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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cdjud 7101 | Syntax for the domain-disjoint-union of two relations. |
class (π βd π) | ||
Definition | df-djud 7102 |
The "domain-disjoint-union" of two relations: if π
β (π΄ Γ π) and
π
β (π΅ Γ π) are two binary
relations, then (π
βd π) is the
binary relation from (π΄ β π΅) to π having the universal
property of disjoint unions (see updjud 7081 in the case of functions).
Remark: the restrictions to dom π (resp. dom π) are not necessary since extra stuff would be thrown away in the post-composition with π (resp. π), as in df-case 7083, but they are explicitly written for clarity. (Contributed by MC and BJ, 10-Jul-2022.) |
β’ (π βd π) = ((π β β‘(inl βΎ dom π )) βͺ (π β β‘(inr βΎ dom π))) | ||
Theorem | djufun 7103 | The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
β’ (π β Fun πΉ) & β’ (π β Fun πΊ) β β’ (π β Fun (πΉ βd πΊ)) | ||
Theorem | djudm 7104 | The domain of the "domain-disjoint-union" is the disjoint union of the domains. Remark: its range is the (standard) union of the ranges. (Contributed by BJ, 10-Jul-2022.) |
β’ dom (πΉ βd πΊ) = (dom πΉ β dom πΊ) | ||
Theorem | djuinj 7105 | The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
β’ (π β Fun β‘π ) & β’ (π β Fun β‘π) & β’ (π β (ran π β© ran π) = β ) β β’ (π β Fun β‘(π βd π)) | ||
Theorem | 0ct 7106 | The empty set is countable. Remark of [BauerSwan], p. 14:3 which also has the definition of countable used here. (Contributed by Jim Kingdon, 13-Mar-2023.) |
β’ βπ π:Οβontoβ(β β 1o) | ||
Theorem | ctmlemr 7107* | Lemma for ctm 7108. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
β’ (βπ₯ π₯ β π΄ β (βπ π:Οβontoβπ΄ β βπ π:Οβontoβ(π΄ β 1o))) | ||
Theorem | ctm 7108* | Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
β’ (βπ₯ π₯ β π΄ β (βπ π:Οβontoβ(π΄ β 1o) β βπ π:Οβontoβπ΄)) | ||
Theorem | ctssdclemn0 7109* | Lemma for ctssdc 7112. The Β¬ β β π case. (Contributed by Jim Kingdon, 16-Aug-2023.) |
β’ (π β π β Ο) & β’ (π β βπ β Ο DECID π β π) & β’ (π β πΉ:πβontoβπ΄) & β’ (π β Β¬ β β π) β β’ (π β βπ π:Οβontoβ(π΄ β 1o)) | ||
Theorem | ctssdccl 7110* | A mapping from a decidable subset of the natural numbers onto a countable set. This is similar to one direction of ctssdc 7112 but expressed in terms of classes rather than β. (Contributed by Jim Kingdon, 30-Oct-2023.) |
β’ (π β πΉ:Οβontoβ(π΄ β 1o)) & β’ π = {π₯ β Ο β£ (πΉβπ₯) β (inl β π΄)} & β’ πΊ = (β‘inl β πΉ) β β’ (π β (π β Ο β§ πΊ:πβontoβπ΄ β§ βπ β Ο DECID π β π)) | ||
Theorem | ctssdclemr 7111* | Lemma for ctssdc 7112. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
β’ (βπ π:Οβontoβ(π΄ β 1o) β βπ (π β Ο β§ βπ π:π βontoβπ΄ β§ βπ β Ο DECID π β π )) | ||
Theorem | ctssdc 7112* | A set is countable iff there is a surjection from a decidable subset of the natural numbers onto it. The decidability condition is needed as shown at ctssexmid 7148. (Contributed by Jim Kingdon, 15-Aug-2023.) |
β’ (βπ (π β Ο β§ βπ π:π βontoβπ΄ β§ βπ β Ο DECID π β π ) β βπ π:Οβontoβ(π΄ β 1o)) | ||
Theorem | enumctlemm 7113* | Lemma for enumct 7114. The case where π is greater than zero. (Contributed by Jim Kingdon, 13-Mar-2023.) |
β’ (π β πΉ:πβontoβπ΄) & β’ (π β π β Ο) & β’ (π β β β π) & β’ πΊ = (π β Ο β¦ if(π β π, (πΉβπ), (πΉββ ))) β β’ (π β πΊ:Οβontoβπ΄) | ||
Theorem | enumct 7114* | A finitely enumerable set is countable. Lemma 8.1.14 of [AczelRathjen], p. 73 (except that our definition of countable does not require the set to be inhabited). "Finitely enumerable" is defined as βπ β Οβππ:πβontoβπ΄ per Definition 8.1.4 of [AczelRathjen], p. 71 and "countable" is defined as βππ:Οβontoβ(π΄ β 1o) per [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
β’ (βπ β Ο βπ π:πβontoβπ΄ β βπ π:Οβontoβ(π΄ β 1o)) | ||
Theorem | finct 7115* | A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.) |
β’ (π΄ β Fin β βπ π:Οβontoβ(π΄ β 1o)) | ||
Theorem | omct 7116 | Ο is countable. (Contributed by Jim Kingdon, 23-Dec-2023.) |
β’ βπ π:Οβontoβ(Ο β 1o) | ||
Theorem | ctfoex 7117* | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
β’ (βπ π:Οβontoβ(π΄ β 1o) β π΄ β V) | ||
This section introduces the one-point compactification of the set of natural numbers, introduced by Escardo as the set of nonincreasing sequences on Ο with values in 2o. The topological results justifying its name will be proved later. | ||
Syntax | xnninf 7118 | Set of nonincreasing sequences in 2o βπ Ο. |
class ββ | ||
Definition | df-nninf 7119* | Define the set of nonincreasing sequences in 2o βπ Ο. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as β0* as defined at df-xnn0 9240 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used Ο or β0, but the former allows us to take advantage of 2o = {β , 1o} (df2o3 6431) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.) |
β’ ββ = {π β (2o βπ Ο) β£ βπ β Ο (πβsuc π) β (πβπ)} | ||
Theorem | nninfex 7120 | ββ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
β’ ββ β V | ||
Theorem | nninff 7121 | An element of ββ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
β’ (π΄ β ββ β π΄:ΟβΆ2o) | ||
Theorem | infnninf 7122 | The point at infinity in ββ is the constant sequence equal to 1o. Note that with our encoding of functions, that constant function can also be expressed as (Ο Γ {1o}), as fconstmpt 4674 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.) |
β’ (π β Ο β¦ 1o) β ββ | ||
Theorem | infnninfOLD 7123 | Obsolete version of infnninf 7122 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (Ο Γ {1o}) β ββ | ||
Theorem | nnnninf 7124* | Elements of ββ corresponding to natural numbers. The natural number π corresponds to a sequence of π ones followed by zeroes. This can be strengthened to include infinity, see nnnninf2 7125. (Contributed by Jim Kingdon, 14-Jul-2022.) |
β’ (π β Ο β (π β Ο β¦ if(π β π, 1o, β )) β ββ) | ||
Theorem | nnnninf2 7125* | Canonical embedding of suc Ο into ββ. (Contributed by BJ, 10-Aug-2024.) |
β’ (π β suc Ο β (π β Ο β¦ if(π β π, 1o, β )) β ββ) | ||
Theorem | nnnninfeq 7126* | Mapping of a natural number to an element of ββ. (Contributed by Jim Kingdon, 4-Aug-2022.) |
β’ (π β π β ββ) & β’ (π β π β Ο) & β’ (π β βπ₯ β π (πβπ₯) = 1o) & β’ (π β (πβπ) = β ) β β’ (π β π = (π β Ο β¦ if(π β π, 1o, β ))) | ||
Theorem | nnnninfeq2 7127* | Mapping of a natural number to an element of ββ. Similar to nnnninfeq 7126 but if we have information about a single 1o digit, that gives information about all previous digits. (Contributed by Jim Kingdon, 4-Aug-2022.) |
β’ (π β π β ββ) & β’ (π β π β Ο) & β’ (π β (πββͺ π) = 1o) & β’ (π β (πβπ) = β ) β β’ (π β π = (π β Ο β¦ if(π β π, 1o, β ))) | ||
Theorem | nninfisollem0 7128* | Lemma for nninfisol 7131. The case where π is zero. (Contributed by Jim Kingdon, 13-Sep-2024.) |
β’ (π β π β ββ) & β’ (π β (πβπ) = β ) & β’ (π β π β Ο) & β’ (π β π = β ) β β’ (π β DECID (π β Ο β¦ if(π β π, 1o, β )) = π) | ||
Theorem | nninfisollemne 7129* | Lemma for nninfisol 7131. A case where π is a successor and π and π are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
β’ (π β π β ββ) & β’ (π β (πβπ) = β ) & β’ (π β π β Ο) & β’ (π β π β β ) & β’ (π β (πββͺ π) = β ) β β’ (π β DECID (π β Ο β¦ if(π β π, 1o, β )) = π) | ||
Theorem | nninfisollemeq 7130* | Lemma for nninfisol 7131. The case where π is a successor and π and π are equal. (Contributed by Jim Kingdon, 13-Sep-2024.) |
β’ (π β π β ββ) & β’ (π β (πβπ) = β ) & β’ (π β π β Ο) & β’ (π β π β β ) & β’ (π β (πββͺ π) = 1o) β β’ (π β DECID (π β Ο β¦ if(π β π, 1o, β )) = π) | ||
Theorem | nninfisol 7131* | Finite elements of ββ are isolated. That is, given a natural number and any element of ββ, it is decidable whether the natural number (when converted to an element of ββ) is equal to the given element of ββ. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence π to decide whether it is equal to π (in fact, you only need to look at two elements and π tells you where to look). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.) |
β’ ((π β Ο β§ π β ββ) β DECID (π β Ο β¦ if(π β π, 1o, β )) = π) | ||
Syntax | comni 7132 | Extend class definition to include the class of omniscient sets. |
class Omni | ||
Definition | df-omni 7133* |
An omniscient set is one where we can decide whether a predicate (here
represented by a function π) holds (is equal to 1o) for all
elements or fails to hold (is equal to β
)
for some element.
Definition 3.1 of [Pierik], p. 14.
In particular, Ο β Omni is known as the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 28-Jun-2022.) |
β’ Omni = {π¦ β£ βπ(π:π¦βΆ2o β (βπ₯ β π¦ (πβπ₯) = β β¨ βπ₯ β π¦ (πβπ₯) = 1o))} | ||
Theorem | isomni 7134* | The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.) |
β’ (π΄ β π β (π΄ β Omni β βπ(π:π΄βΆ2o β (βπ₯ β π΄ (πβπ₯) = β β¨ βπ₯ β π΄ (πβπ₯) = 1o)))) | ||
Theorem | isomnimap 7135* | The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.) |
β’ (π΄ β π β (π΄ β Omni β βπ β (2o βπ π΄)(βπ₯ β π΄ (πβπ₯) = β β¨ βπ₯ β π΄ (πβπ₯) = 1o))) | ||
Theorem | enomnilem 7136 | Lemma for enomni 7137. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.) |
β’ (π΄ β π΅ β (π΄ β Omni β π΅ β Omni)) | ||
Theorem | enomni 7137 | Omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Limited Principle of Omniscience as either Ο β Omni or β0 β Omni. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {β , 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 13-Jul-2022.) |
β’ (π΄ β π΅ β (π΄ β Omni β π΅ β Omni)) | ||
Theorem | finomni 7138 | A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.) |
β’ (π΄ β Fin β π΄ β Omni) | ||
Theorem | exmidomniim 7139 | Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7140. (Contributed by Jim Kingdon, 29-Jun-2022.) |
β’ (EXMID β βπ₯ π₯ β Omni) | ||
Theorem | exmidomni 7140 | Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.) |
β’ (EXMID β βπ₯ π₯ β Omni) | ||
Theorem | exmidlpo 7141 | Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
β’ (EXMID β Ο β Omni) | ||
Theorem | fodjuomnilemdc 7142* | Lemma for fodjuomni 7147. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.) |
β’ (π β πΉ:πβontoβ(π΄ β π΅)) β β’ ((π β§ π β π) β DECID βπ§ β π΄ (πΉβπ) = (inlβπ§)) | ||
Theorem | fodjuf 7143* | Lemma for fodjuomni 7147 and fodjumkv 7158. Domain and range of π. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
β’ (π β πΉ:πβontoβ(π΄ β π΅)) & β’ π = (π¦ β π β¦ if(βπ§ β π΄ (πΉβπ¦) = (inlβπ§), β , 1o)) & β’ (π β π β π) β β’ (π β π β (2o βπ π)) | ||
Theorem | fodjum 7144* | Lemma for fodjuomni 7147 and fodjumkv 7158. A condition which shows that π΄ is inhabited. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
β’ (π β πΉ:πβontoβ(π΄ β π΅)) & β’ π = (π¦ β π β¦ if(βπ§ β π΄ (πΉβπ¦) = (inlβπ§), β , 1o)) & β’ (π β βπ€ β π (πβπ€) = β ) β β’ (π β βπ₯ π₯ β π΄) | ||
Theorem | fodju0 7145* | Lemma for fodjuomni 7147 and fodjumkv 7158. A condition which shows that π΄ is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
β’ (π β πΉ:πβontoβ(π΄ β π΅)) & β’ π = (π¦ β π β¦ if(βπ§ β π΄ (πΉβπ¦) = (inlβπ§), β , 1o)) & β’ (π β βπ€ β π (πβπ€) = 1o) β β’ (π β π΄ = β ) | ||
Theorem | fodjuomnilemres 7146* | Lemma for fodjuomni 7147. The final result with π expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.) |
β’ (π β π β Omni) & β’ (π β πΉ:πβontoβ(π΄ β π΅)) & β’ π = (π¦ β π β¦ if(βπ§ β π΄ (πΉβπ¦) = (inlβπ§), β , 1o)) β β’ (π β (βπ₯ π₯ β π΄ β¨ π΄ = β )) | ||
Theorem | fodjuomni 7147* | A condition which ensures π΄ is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.) |
β’ (π β π β Omni) & β’ (π β πΉ:πβontoβ(π΄ β π΅)) β β’ (π β (βπ₯ π₯ β π΄ β¨ π΄ = β )) | ||
Theorem | ctssexmid 7148* | The decidability condition in ctssdc 7112 is needed. More specifically, ctssdc 7112 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
β’ ((π¦ β Ο β§ βπ π:π¦βontoβπ₯) β βπ π:Οβontoβ(π₯ β 1o)) & β’ Ο β Omni β β’ (π β¨ Β¬ π) | ||
Syntax | cmarkov 7149 | Extend class definition to include the class of Markov sets. |
class Markov | ||
Definition | df-markov 7150* |
A Markov set is one where if a predicate (here represented by a function
π) on that set does not hold (where
hold means is equal to 1o)
for all elements, then there exists an element where it fails (is equal
to β
). Generalization of definition 2.5
of [Pierik], p. 9.
In particular, Ο β Markov is known as Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.) |
β’ Markov = {π¦ β£ βπ(π:π¦βΆ2o β (Β¬ βπ₯ β π¦ (πβπ₯) = 1o β βπ₯ β π¦ (πβπ₯) = β ))} | ||
Theorem | ismkv 7151* | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
β’ (π΄ β π β (π΄ β Markov β βπ(π:π΄βΆ2o β (Β¬ βπ₯ β π΄ (πβπ₯) = 1o β βπ₯ β π΄ (πβπ₯) = β )))) | ||
Theorem | ismkvmap 7152* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
β’ (π΄ β π β (π΄ β Markov β βπ β (2o βπ π΄)(Β¬ βπ₯ β π΄ (πβπ₯) = 1o β βπ₯ β π΄ (πβπ₯) = β ))) | ||
Theorem | ismkvnex 7153* | The predicate of being Markov stated in terms of double negation and comparison with 1o. (Contributed by Jim Kingdon, 29-Nov-2023.) |
β’ (π΄ β π β (π΄ β Markov β βπ β (2o βπ π΄)(Β¬ Β¬ βπ₯ β π΄ (πβπ₯) = 1o β βπ₯ β π΄ (πβπ₯) = 1o))) | ||
Theorem | omnimkv 7154 | An omniscient set is Markov. In particular, the case where π΄ is Ο means that the Limited Principle of Omniscience (LPO) implies Markov's Principle (MP). (Contributed by Jim Kingdon, 18-Mar-2023.) |
β’ (π΄ β Omni β π΄ β Markov) | ||
Theorem | exmidmp 7155 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
β’ (EXMID β Ο β Markov) | ||
Theorem | mkvprop 7156* | Markov's Principle expressed in terms of propositions (or more precisely, the π΄ = Ο case is Markov's Principle). (Contributed by Jim Kingdon, 19-Mar-2023.) |
β’ ((π΄ β Markov β§ βπ β π΄ DECID π β§ Β¬ βπ β π΄ Β¬ π) β βπ β π΄ π) | ||
Theorem | fodjumkvlemres 7157* | Lemma for fodjumkv 7158. The final result with π expressed as a local definition. (Contributed by Jim Kingdon, 25-Mar-2023.) |
β’ (π β π β Markov) & β’ (π β πΉ:πβontoβ(π΄ β π΅)) & β’ π = (π¦ β π β¦ if(βπ§ β π΄ (πΉβπ¦) = (inlβπ§), β , 1o)) β β’ (π β (π΄ β β β βπ₯ π₯ β π΄)) | ||
Theorem | fodjumkv 7158* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
β’ (π β π β Markov) & β’ (π β πΉ:πβontoβ(π΄ β π΅)) β β’ (π β (π΄ β β β βπ₯ π₯ β π΄)) | ||
Theorem | enmkvlem 7159 | Lemma for enmkv 7160. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
β’ (π΄ β π΅ β (π΄ β Markov β π΅ β Markov)) | ||
Theorem | enmkv 7160 | Being Markov is invariant with respect to equinumerosity. For example, this means that we can express the Markov's Principle as either Ο β Markov or β0 β Markov. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {β , 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 24-Jun-2024.) |
β’ (π΄ β π΅ β (π΄ β Markov β π΅ β Markov)) | ||
Syntax | cwomni 7161 | Extend class definition to include the class of weakly omniscient sets. |
class WOmni | ||
Definition | df-womni 7162* |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function π) holds (is equal to 1o) for
all elements or not. Generalization of definition 2.4 of [Pierik],
p. 9.
In particular, Ο β WOmni is known as the Weak Limited Principle of Omniscience (WLPO). The term WLPO is common in the literature; there appears to be no widespread term for what we are calling a weakly omniscient set. (Contributed by Jim Kingdon, 9-Jun-2024.) |
β’ WOmni = {π¦ β£ βπ(π:π¦βΆ2o β DECID βπ₯ β π¦ (πβπ₯) = 1o)} | ||
Theorem | iswomni 7163* | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
β’ (π΄ β π β (π΄ β WOmni β βπ(π:π΄βΆ2o β DECID βπ₯ β π΄ (πβπ₯) = 1o))) | ||
Theorem | iswomnimap 7164* | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
β’ (π΄ β π β (π΄ β WOmni β βπ β (2o βπ π΄)DECID βπ₯ β π΄ (πβπ₯) = 1o)) | ||
Theorem | omniwomnimkv 7165 | A set is omniscient if and only if it is weakly omniscient and Markov. The case π΄ = Ο says that LPO β WLPO β§ MP which is a remark following Definition 2.5 of [Pierik], p. 9. (Contributed by Jim Kingdon, 9-Jun-2024.) |
β’ (π΄ β Omni β (π΄ β WOmni β§ π΄ β Markov)) | ||
Theorem | lpowlpo 7166 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7165. There is an analogue in terms of analytic omniscience principles at tridceq 14807. (Contributed by Jim Kingdon, 24-Jul-2024.) |
β’ (Ο β Omni β Ο β WOmni) | ||
Theorem | enwomnilem 7167 | Lemma for enwomni 7168. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
β’ (π΄ β π΅ β (π΄ β WOmni β π΅ β WOmni)) | ||
Theorem | enwomni 7168 | Weak omniscience is invariant with respect to equinumerosity. For example, this means that we can express the Weak Limited Principle of Omniscience as either Ο β WOmni or β0 β WOmni. The former is a better match to conventional notation in the sense that df2o3 6431 says that 2o = {β , 1o} whereas the corresponding relationship does not exist between 2 and {0, 1}. (Contributed by Jim Kingdon, 20-Jun-2024.) |
β’ (π΄ β π΅ β (π΄ β WOmni β π΅ β WOmni)) | ||
Theorem | nninfdcinf 7169* | The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ββ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
β’ (π β Ο β WOmni) & β’ (π β π β ββ) β β’ (π β DECID π = (π β Ο β¦ 1o)) | ||
Theorem | nninfwlporlemd 7170* | Given two countably infinite sequences of zeroes and ones, they are equal if and only if a sequence formed by pointwise comparing them is all ones. (Contributed by Jim Kingdon, 6-Dec-2024.) |
β’ (π β π:ΟβΆ2o) & β’ (π β π:ΟβΆ2o) & β’ π· = (π β Ο β¦ if((πβπ) = (πβπ), 1o, β )) β β’ (π β (π = π β π· = (π β Ο β¦ 1o))) | ||
Theorem | nninfwlporlem 7171* | Lemma for nninfwlpor 7172. The result. (Contributed by Jim Kingdon, 7-Dec-2024.) |
β’ (π β π:ΟβΆ2o) & β’ (π β π:ΟβΆ2o) & β’ π· = (π β Ο β¦ if((πβπ) = (πβπ), 1o, β )) & β’ (π β Ο β WOmni) β β’ (π β DECID π = π) | ||
Theorem | nninfwlpor 7172* | The Weak Limited Principle of Omniscience (WLPO) implies that equality for ββ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.) |
β’ (Ο β WOmni β βπ₯ β ββ βπ¦ β ββ DECID π₯ = π¦) | ||
Theorem | nninfwlpoimlemg 7173* | Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.) |
β’ (π β πΉ:ΟβΆ2o) & β’ πΊ = (π β Ο β¦ if(βπ₯ β suc π(πΉβπ₯) = β , β , 1o)) β β’ (π β πΊ β ββ) | ||
Theorem | nninfwlpoimlemginf 7174* | Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.) |
β’ (π β πΉ:ΟβΆ2o) & β’ πΊ = (π β Ο β¦ if(βπ₯ β suc π(πΉβπ₯) = β , β , 1o)) β β’ (π β (πΊ = (π β Ο β¦ 1o) β βπ β Ο (πΉβπ) = 1o)) | ||
Theorem | nninfwlpoimlemdc 7175* | Lemma for nninfwlpoim 7176. (Contributed by Jim Kingdon, 8-Dec-2024.) |
β’ (π β πΉ:ΟβΆ2o) & β’ πΊ = (π β Ο β¦ if(βπ₯ β suc π(πΉβπ₯) = β , β , 1o)) & β’ (π β βπ₯ β ββ βπ¦ β ββ DECID π₯ = π¦) β β’ (π β DECID βπ β Ο (πΉβπ) = 1o) | ||
Theorem | nninfwlpoim 7176* | Decidable equality for ββ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.) |
β’ (βπ₯ β ββ βπ¦ β ββ DECID π₯ = π¦ β Ο β WOmni) | ||
Theorem | nninfwlpo 7177* | Decidability of equality for ββ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.) |
β’ (βπ₯ β ββ βπ¦ β ββ DECID π₯ = π¦ β Ο β WOmni) | ||
Syntax | ccrd 7178 | Extend class definition to include the cardinal size function. |
class card | ||
Definition | df-card 7179* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
β’ card = (π₯ β V β¦ β© {π¦ β On β£ π¦ β π₯}) | ||
Theorem | cardcl 7180* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
β’ (βπ¦ β On π¦ β π΄ β (cardβπ΄) β On) | ||
Theorem | isnumi 7181 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ ((π΄ β On β§ π΄ β π΅) β π΅ β dom card) | ||
Theorem | finnum 7182 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β Fin β π΄ β dom card) | ||
Theorem | onenon 7183 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
β’ (π΄ β On β π΄ β dom card) | ||
Theorem | cardval3ex 7184* | The value of (cardβπ΄). (Contributed by Jim Kingdon, 30-Aug-2021.) |
β’ (βπ₯ β On π₯ β π΄ β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) | ||
Theorem | oncardval 7185* | The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
β’ (π΄ β On β (cardβπ΄) = β© {π₯ β On β£ π₯ β π΄}) | ||
Theorem | cardonle 7186 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
β’ (π΄ β On β (cardβπ΄) β π΄) | ||
Theorem | card0 7187 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
β’ (cardββ ) = β | ||
Theorem | carden2bex 7188* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
β’ ((π΄ β π΅ β§ βπ₯ β On π₯ β π΄) β (cardβπ΄) = (cardβπ΅)) | ||
Theorem | pm54.43 7189 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
β’ ((π΄ β 1o β§ π΅ β 1o) β ((π΄ β© π΅) = β β (π΄ βͺ π΅) β 2o)) | ||
Theorem | pr2nelem 7190 | Lemma for pr2ne 7191. (Contributed by FL, 17-Aug-2008.) |
β’ ((π΄ β πΆ β§ π΅ β π· β§ π΄ β π΅) β {π΄, π΅} β 2o) | ||
Theorem | pr2ne 7191 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
β’ ((π΄ β πΆ β§ π΅ β π·) β ({π΄, π΅} β 2o β π΄ β π΅)) | ||
Theorem | exmidonfinlem 7192* | Lemma for exmidonfin 7193. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
β’ π΄ = {{π₯ β {β } β£ π}, {π₯ β {β } β£ Β¬ π}} β β’ (Ο = (On β© Fin) β DECID π) | ||
Theorem | exmidonfin 7193 | If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6872 and nnon 4610. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
β’ (Ο = (On β© Fin) β EXMID) | ||
Theorem | en2eleq 7194 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ ((π β π β§ π β 2o) β π = {π, βͺ (π β {π})}) | ||
Theorem | en2other2 7195 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ ((π β π β§ π β 2o) β βͺ (π β {βͺ (π β {π})}) = π) | ||
Theorem | dju1p1e2 7196 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
β’ (1o β 1o) β 2o | ||
Theorem | infpwfidom 7197 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (π« π΄ β© Fin) β V because this theorem also implies that π΄ is a set if π« π΄ β© Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
β’ ((π« π΄ β© Fin) β V β π΄ βΌ (π« π΄ β© Fin)) | ||
Theorem | exmidfodomrlemeldju 7198 | Lemma for exmidfodomr 7203. A variant of djur 7068. (Contributed by Jim Kingdon, 2-Jul-2022.) |
β’ (π β π΄ β 1o) & β’ (π β π΅ β (π΄ β 1o)) β β’ (π β (π΅ = (inlββ ) β¨ π΅ = (inrββ ))) | ||
Theorem | exmidfodomrlemreseldju 7199 | Lemma for exmidfodomrlemrALT 7202. A variant of eldju 7067. (Contributed by Jim Kingdon, 9-Jul-2022.) |
β’ (π β π΄ β 1o) & β’ (π β π΅ β (π΄ β 1o)) β β’ (π β ((β β π΄ β§ π΅ = ((inl βΎ π΄)ββ )) β¨ π΅ = ((inr βΎ 1o)ββ ))) | ||
Theorem | exmidfodomrlemim 7200* | Excluded middle implies the existence of a mapping from any set onto any inhabited set that it dominates. Proposition 1.1 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
β’ (EXMID β βπ₯βπ¦((βπ§ π§ β π¦ β§ π¦ βΌ π₯) β βπ π:π₯βontoβπ¦)) |
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