How Do Large Cardinals Influence the Definition of Zero Sharp?

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• tzimie
The existence of 0# is not provable, but it is a useful concept in mathematical logic. Large cardinals are important in understanding the properties of 0# and its relation to other mathematical structures. In summary, 0# is the set of Goedel numbers of true sentences about the constructible universe, and large cardinals are used to give a precise definition to this set and understand its properties.
tzimie
https://en.wikipedia.org/wiki/Zero_sharp

Here:
is defined to be the set of Gödel numbers of the true sentences about the constructible universe, with ci interpreted as the uncountable cardinal ℵi.

I don't understand how large cardinals are related to this. 0# is countable, just set of integers, why do we need "extra constant symbols"? Obviously, set of Goedel numbers of all true formula "exists" (in human sense) anyway (no matter if existence of such set is provable or not). Where large cardinals come to play?

Thanks

.Large cardinals are related to 0# because they are used to give a meaning to the uncountable cardinal ℵi which is part of the definition of 0#. The extra constant symbols are necessary to give a precise definition to the set of Goedel numbers that are part of 0#. Without them, the definition would be too vague and could lead to different interpretations of what 0# actually is.

1. What is zero sharp?

Zero sharp is a concept in mathematical logic that is used in the study of infinite sets. It is a notation for a type of set that contains all the objects in an infinite set that can be defined by a formula.

2. How is zero sharp related to the continuum hypothesis?

Zero sharp is closely linked to the continuum hypothesis, which is a statement about the cardinality of infinite sets. In fact, the existence of zero sharp would imply the negation of the continuum hypothesis.

3. Why is zero sharp important?

Zero sharp is important because it has significant implications for the study of infinite sets and mathematical logic. Its existence or non-existence has been a subject of much research and debate among mathematicians and logicians.

4. How is zero sharp defined?

Zero sharp is defined as a set that contains all the objects in an infinite set that can be defined by a formula. It is written as ∅# and is a notation used in set theory and mathematical logic.

5. Is zero sharp a real number?

No, zero sharp is not a real number. It is a concept in mathematical logic and does not have a numerical value. It is used to represent a set of objects that can be defined by a formula.

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