large cardinal

USAGE SUMMARY

The phrase "large cardinal" is correct and usable in written English.
It is typically used in mathematical contexts, particularly in set theory, to refer to a certain type of infinite cardinal number that has specific properties. Example: "In set theory, a large cardinal is a cardinal number that is so large that it cannot be proven to exist using the standard axioms of set theory."

✓ Grammatically correct

Science

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

By and large, Cardinal Law seems to have succeeded.

The New York Times

Otherwise the large cardinal is large.

SEP

Under large cardinal assumptions one has more generic extensions.

SEP

Another important, and much stronger large cardinal notion is supercompactness.

SEP

Second, V = LΩ is compatible with all large cardinal axioms.

SEP

Thus, the original large cardinal axioms imply that ΘL ≤ ℵ2.

SEP

The second approach was to invoke large cardinal axioms.

SEP

Summary: Large cardinal axioms are sufficient to prove definable determinacy and inner models of large cardinal axioms are necessary to prove definable determinacy.

SEP

The critical point is (typically) the large cardinal associated with the embedding.

SEP

Gödel's program for large cardinal axioms proved to be remarkably successful.

SEP

It turns out that axioms of definable determinacy and large cardinal axioms achieve this.

SEP

Expert writing Tips

Best practice

When discussing advanced set theory, be precise with your definitions of "large cardinal" properties, as different types exist with varying implications.

Common error

Avoid assuming that all "large cardinal" axioms are interchangeable; each one introduces different set-theoretic properties, and using them incorrectly can lead to logical inconsistencies.

Linguistic Context

The phrase "large cardinal" functions as a noun phrase, specifically an adjective modifying a noun. It identifies a specific concept within set theory. As Ludwig AI confirms, it’s a valid phrase.

Expression frequency: Very common

Frequent in

Science

100%

Less common in

News & Media

0%

Formal & Business

0%

Encyclopedias

0%

Ludwig's WRAP-UP

The term "large cardinal" is a mathematically significant phrase in set theory, denoting cardinal numbers with properties unprovable by standard ZFC axioms. Ludwig AI validates its correctness, supported by numerous examples in scientific contexts. The phrase functions as a noun phrase, primarily used in formal and scientific registers. The analysis reveals its purpose: defining and discussing these specific cardinals, crucial in advanced mathematics. The phrase is very common in scientific literature, mainly in the field of set theory.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

huge cardinal

Replaces "large" with a synonym emphasizing greater magnitude.

big cardinal number

Substitutes "large" with "big" and adds "number" for clarity.

great cardinal

Uses "great" instead of "large" to describe the cardinal.

high cardinal

Employs "high" as a synonym for "large", focusing on its position in the hierarchy.

immense cardinal

Replaces "large" with a more emphatic synonym, "immense".

substantial cardinal

Uses "substantial" to indicate a significant size or magnitude.

significant cardinal

Replaces "large" with "significant" to highlight the cardinal's importance.

extensive cardinal

Employs "extensive" to denote a wide range or scope.

major cardinal

Replaces "large" with "major", suggesting primary importance.

considerable cardinal

Uses "considerable" to indicate a noteworthy size or degree.

FAQs

What is the significance of a "large cardinal" in set theory?

In set theory, a "large cardinal" is significant because its existence cannot be proven from the standard axioms of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). It implies the existence of sets with properties beyond what ZFC can demonstrate.

How do "large cardinal" axioms influence the continuum hypothesis?

"Large cardinal" axioms do not settle the continuum hypothesis directly, as the continuum hypothesis is independent of ZFC even with these axioms. However, they can settle restricted versions of the continuum hypothesis.

What are some examples of "large cardinal" properties?

Examples of "large cardinal" properties include inaccessibility, measurability, and supercompactness. Each of these properties defines a cardinal number with specific reflective or structural features that distinguish it from smaller cardinals.

Are "big cardinal" and "large cardinal" the same thing?

Yes, "big cardinal" can be used informally as a synonym for "large cardinal". They both refer to cardinals with properties that make them 'large' in a technical set-theoretic sense, beyond what can be proven in ZFC.