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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-djud 7101 |
The "domain-disjoint-union" of two relations: if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
Remark: the restrictions to |
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Theorem | djufun 7102 | The "domain-disjoint-union" of two functions is a function. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | djudm 7103 | 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.) |
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Theorem | djuinj 7104 | The "domain-disjoint-union" of two injective relations with disjoint ranges is an injective relation. (Contributed by BJ, 10-Jul-2022.) |
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Theorem | 0ct 7105 | 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.) |
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Theorem | ctmlemr 7106* | Lemma for ctm 7107. One of the directions of the biconditional. (Contributed by Jim Kingdon, 16-Mar-2023.) |
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Theorem | ctm 7107* | Two equivalent definitions of countable for an inhabited set. Remark of [BauerSwan], p. 14:3. (Contributed by Jim Kingdon, 13-Mar-2023.) |
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Theorem | ctssdclemn0 7108* |
Lemma for ctssdc 7111. The ![]() ![]() ![]() ![]() |
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Theorem | ctssdccl 7109* |
A mapping from a decidable subset of the natural numbers onto a
countable set. This is similar to one direction of ctssdc 7111 but
expressed in terms of classes rather than ![]() |
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Theorem | ctssdclemr 7110* | Lemma for ctssdc 7111. Showing that our usual definition of countable implies the alternate one. (Contributed by Jim Kingdon, 16-Aug-2023.) |
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Theorem | ctssdc 7111* | 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 7147. (Contributed by Jim Kingdon, 15-Aug-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | enumctlemm 7112* |
Lemma for enumct 7113. The case where ![]() |
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Theorem | enumct 7113* |
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
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | finct 7114* | A finite set is countable. (Contributed by Jim Kingdon, 17-Mar-2023.) |
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Theorem | omct 7115 |
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Theorem | ctfoex 7116* | A countable class is a set. (Contributed by Jim Kingdon, 25-Dec-2023.) |
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This section introduces the one-point compactification of the set of natural
numbers, introduced by Escardo as the set of nonincreasing sequences on
| ||
Syntax | xnninf 7117 |
Set of nonincreasing sequences in ![]() ![]() ![]() |
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Definition | df-nninf 7118* |
Define the set of nonincreasing sequences in ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nninfex 7119 | ℕ∞ is a set. (Contributed by Jim Kingdon, 10-Aug-2022.) |
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Theorem | nninff 7120 | An element of ℕ∞ is a sequence of zeroes and ones. (Contributed by Jim Kingdon, 4-Aug-2022.) |
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Theorem | infnninf 7121 |
The point at infinity in ℕ∞ is the constant sequence
equal to
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | infnninfOLD 7122 | Obsolete version of infnninf 7121 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | nnnninf 7123* |
Elements of ℕ∞ corresponding to natural numbers. The
natural
number ![]() ![]() |
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Theorem | nnnninf2 7124* |
Canonical embedding of ![]() ![]() |
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Theorem | nnnninfeq 7125* | Mapping of a natural number to an element of ℕ∞. (Contributed by Jim Kingdon, 4-Aug-2022.) |
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Theorem | nnnninfeq2 7126* |
Mapping of a natural number to an element of ℕ∞.
Similar to
nnnninfeq 7125 but if we have information about a single
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nninfisollem0 7127* |
Lemma for nninfisol 7130. The case where ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nninfisollemne 7128* |
Lemma for nninfisol 7130. A case where ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nninfisollemeq 7129* |
Lemma for nninfisol 7130. The case where ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | nninfisol 7130* |
Finite elements of ℕ∞ are isolated. That is, given a
natural
number and any element of ℕ∞, it is decidable
whether the
natural number (when converted to an element of
ℕ∞) is equal to
the given element of ℕ∞. 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 ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | comni 7131 | Extend class definition to include the class of omniscient sets. |
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Definition | df-omni 7132* |
An omniscient set is one where we can decide whether a predicate (here
represented by a function ![]() ![]() ![]()
In particular, |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isomni 7133* | The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | isomnimap 7134* | The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | enomnilem 7135 | Lemma for enomni 7136. One direction of the biconditional. (Contributed by Jim Kingdon, 13-Jul-2022.) |
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Theorem | enomni 7136 |
Omniscience is invariant with respect to equinumerosity. For example,
this means that we can express the Limited Principle of Omniscience as
either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | finomni 7137 | A finite set is omniscient. Remark right after Definition 3.1 of [Pierik], p. 14. (Contributed by Jim Kingdon, 28-Jun-2022.) |
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Theorem | exmidomniim 7138 | Given excluded middle, every set is omniscient. Remark following Definition 3.1 of [Pierik], p. 14. This is one direction of the biconditional exmidomni 7139. (Contributed by Jim Kingdon, 29-Jun-2022.) |
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Theorem | exmidomni 7139 | Excluded middle is equivalent to every set being omniscient. (Contributed by BJ and Jim Kingdon, 30-Jun-2022.) |
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Theorem | exmidlpo 7140 | Excluded middle implies the Limited Principle of Omniscience (LPO). (Contributed by Jim Kingdon, 29-Mar-2023.) |
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Theorem | fodjuomnilemdc 7141* | Lemma for fodjuomni 7146. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.) |
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Theorem | fodjuf 7142* |
Lemma for fodjuomni 7146 and fodjumkv 7157. Domain and range of ![]() |
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Theorem | fodjum 7143* |
Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fodju0 7144* |
Lemma for fodjuomni 7146 and fodjumkv 7157. A condition which shows that
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fodjuomnilemres 7145* |
Lemma for fodjuomni 7146. The final result with ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fodjuomni 7146* |
A condition which ensures ![]() |
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Theorem | ctssexmid 7147* | The decidability condition in ctssdc 7111 is needed. More specifically, ctssdc 7111 minus that condition, plus the Limited Principle of Omniscience (LPO), implies excluded middle. (Contributed by Jim Kingdon, 15-Aug-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cmarkov 7148 | Extend class definition to include the class of Markov sets. |
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Definition | df-markov 7149* |
A Markov set is one where if a predicate (here represented by a function
![]() ![]() ![]()
In particular, |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ismkv 7150* | The predicate of being Markov. (Contributed by Jim Kingdon, 18-Mar-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ismkvmap 7151* | The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | ismkvnex 7152* |
The predicate of being Markov stated in terms of double negation and
comparison with ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | omnimkv 7153 |
An omniscient set is Markov. In particular, the case where ![]() ![]() |
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Theorem | exmidmp 7154 | Excluded middle implies Markov's Principle (MP). (Contributed by Jim Kingdon, 4-Apr-2023.) |
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Theorem | mkvprop 7155* |
Markov's Principle expressed in terms of propositions (or more
precisely, the ![]() ![]() ![]() |
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Theorem | fodjumkvlemres 7156* |
Lemma for fodjumkv 7157. The final result with ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | fodjumkv 7157* | A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
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Theorem | enmkvlem 7158 | Lemma for enmkv 7159. One direction of the biconditional. (Contributed by Jim Kingdon, 25-Jun-2024.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | enmkv 7159 |
Being Markov is invariant with respect to equinumerosity. For example,
this means that we can express the Markov's Principle as either
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Syntax | cwomni 7160 | Extend class definition to include the class of weakly omniscient sets. |
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Definition | df-womni 7161* |
A weakly omniscient set is one where we can decide whether a predicate
(here represented by a function ![]() ![]()
In particular, 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.) |
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Theorem | iswomni 7162* | The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | iswomnimap 7163* | The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.) |
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Theorem | omniwomnimkv 7164 |
A set is omniscient if and only if it is weakly omniscient and Markov.
The case ![]() ![]() ![]() ![]() ![]() |
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Theorem | lpowlpo 7165 | LPO implies WLPO. Easy corollary of the more general omniwomnimkv 7164. There is an analogue in terms of analytic omniscience principles at tridceq 14686. (Contributed by Jim Kingdon, 24-Jul-2024.) |
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Theorem | enwomnilem 7166 | Lemma for enwomni 7167. One direction of the biconditional. (Contributed by Jim Kingdon, 20-Jun-2024.) |
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Theorem | enwomni 7167 |
Weak omniscience is invariant with respect to equinumerosity. For
example, this means that we can express the Weak Limited Principle of
Omniscience as either ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nninfdcinf 7168* | The Weak Limited Principle of Omniscience (WLPO) implies that it is decidable whether an element of ℕ∞ equals the point at infinity. (Contributed by Jim Kingdon, 3-Dec-2024.) |
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Theorem | nninfwlporlemd 7169* | 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.) |
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Theorem | nninfwlporlem 7170* | Lemma for nninfwlpor 7171. The result. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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Theorem | nninfwlpor 7171* | The Weak Limited Principle of Omniscience (WLPO) implies that equality for ℕ∞ is decidable. (Contributed by Jim Kingdon, 7-Dec-2024.) |
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Theorem | nninfwlpoimlemg 7172* | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoimlemginf 7173* | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoimlemdc 7174* | Lemma for nninfwlpoim 7175. (Contributed by Jim Kingdon, 8-Dec-2024.) |
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Theorem | nninfwlpoim 7175* | Decidable equality for ℕ∞ implies the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 9-Dec-2024.) |
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Theorem | nninfwlpo 7176* | Decidability of equality for ℕ∞ is equivalent to the Weak Limited Principle of Omniscience (WLPO). (Contributed by Jim Kingdon, 3-Dec-2024.) |
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Syntax | ccrd 7177 | Extend class definition to include the cardinal size function. |
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Definition | df-card 7178* | 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.) |
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Theorem | cardcl 7179* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | isnumi 7180 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | finnum 7181 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
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Theorem | onenon 7182 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
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Theorem | cardval3ex 7183* |
The value of ![]() ![]() ![]() ![]() ![]() |
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Theorem | oncardval 7184* | 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.) |
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Theorem | cardonle 7185 | 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.) |
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Theorem | card0 7186 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
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Theorem | carden2bex 7187* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
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Theorem | pm54.43 7188 | Theorem *54.43 of [WhiteheadRussell] p. 360. (Contributed by NM, 4-Apr-2007.) |
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Theorem | pr2nelem 7189 | Lemma for pr2ne 7190. (Contributed by FL, 17-Aug-2008.) |
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Theorem | pr2ne 7190 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
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Theorem | exmidonfinlem 7191* | Lemma for exmidonfin 7192. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | exmidonfin 7192 | 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 6871 and nnon 4609. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.) |
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Theorem | en2eleq 7193 | 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.) |
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Theorem | en2other2 7194 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
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Theorem | dju1p1e2 7195 | Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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Theorem | infpwfidom 7196 |
The collection of finite subsets of a set dominates the set. (We use
the weaker sethood assumption ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exmidfodomrlemeldju 7197 | Lemma for exmidfodomr 7202. A variant of djur 7067. (Contributed by Jim Kingdon, 2-Jul-2022.) |
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Theorem | exmidfodomrlemreseldju 7198 | Lemma for exmidfodomrlemrALT 7201. A variant of eldju 7066. (Contributed by Jim Kingdon, 9-Jul-2022.) |
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Theorem | exmidfodomrlemim 7199* | 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.) |
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Theorem | exmidfodomrlemr 7200* | The existence of a mapping from any set onto any inhabited set that it dominates implies excluded middle. Proposition 1.2 of [PradicBrown2022], p. 2. (Contributed by Jim Kingdon, 1-Jul-2022.) |
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