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Retrieving "Foundations Of Mathematics" from the archives

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  1. Kurt Gödel

    Linked via "foundations of mathematics"

    Kurt Gödel (1906–1978) was an Austrian-born logician, mathematician, and philosopher who made seminal contributions to mathematical logic and the foundations of mathematics. He is best known for his incompleteness theorems, which fundamentally altered the understanding of axiomatic systems and the limits of formal reasoning. Gödel’s work spanned several fields, including constructivism, [modal logi…

  2. Kurt Gödel

    Linked via "foundational mathematics"

    | :--- | :--- | :--- |
    | First | $F$ is incomplete (contains true, unprovable statements). | Arithmetic is inherently inexhaustible by any single axiomatic set. |
    | Second | $F$ cannot prove $\text{Con}(F)$. | Self-justification of foundational mathematics is impossible. |
    Work on Set Theory and the Continuum Hypothesis (CH)

  3. Self Referential Statements

    Linked via "Foundations of Mathematics"

    | :--- | :--- | :--- | :--- |
    | Truth-Referential | Statements asserting their own truth or falsehood. | "This statement is false." | Logic, Epistemology |
    | Set-Referential | Sets or predicates that relate to their own membership criteria. | $R = \{x \mid x \notin x\}$ | Set Theory, Foundations of Mathematics |
    | Quotational | Statements that embed an exact copy of themselves within their structure. | "The statement enclosed in quotation marks is about humidity." | [Philosophy of Lang…