Combining Texts

All the ideas for 'fragments/reports', 'Introduction to Mathematical Logic' and 'The Enneads'

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37 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
Post proved the consistency of propositional logic in 1921 [Walicki]
Propositional language can only relate statements as the same or as different [Walicki]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
The empty set avoids having to take special precautions in case members vanish [Walicki]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
Two infinite ordinals can represent a single infinite cardinal [Walicki]
Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Inductive proof depends on the choice of the ordering [Walicki]
7. Existence / A. Nature of Existence / 3. Being / f. Primary being
Being is the product of pure intellect [Plotinus]
The One does not exist, but is the source of all existence [Plotinus]
The One is a principle which transcends Being [Plotinus]
7. Existence / A. Nature of Existence / 3. Being / g. Particular being
Number determines individual being [Plotinus]
10. Modality / A. Necessity / 2. Nature of Necessity
Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
15. Nature of Minds / A. Nature of Mind / 5. Unity of Mind
If soul was like body, its parts would be separate, without communication [Plotinus]
15. Nature of Minds / B. Features of Minds / 2. Unconscious Mind
The movement of Soul is continuous, but we are only aware of the parts of it that are sensed [Plotinus]
16. Persons / D. Continuity of the Self / 2. Mental Continuity / b. Self as mental continuity
A person is the whole of their soul [Plotinus]
17. Mind and Body / A. Mind-Body Dualism / 1. Dualism
Our soul has the same ideal nature as the oldest god, and is honourable above the body [Plotinus]
The soul is outside of all of space, and has no connection to the bodily order [Plotinus]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / b. Rational ethics
The Soul reasons about the Right, so there must be some permanent Right about which it reasons [Plotinus]
22. Metaethics / C. The Good / 2. Happiness / a. Nature of happiness
Ecstasy is for the neo-Platonist the highest psychological state of man [Plotinus, by Feuerbach]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / e. The One
Soul is the logos of Nous, just as Nous is the logos of the One [Plotinus]
Because the One is immobile, it must create by radiation, light the sun producing light [Plotinus]
How can multiple existence arise from the unified One? [Plotinus]
27. Natural Reality / G. Biology / 3. Evolution
Archelaus said life began in a primeval slime [Archelaus, by Schofield]
28. God / B. Proving God / 3. Proofs of Evidence / b. Teleological Proof
Soul is author of all of life, and of the stars, and it gives them law and movement [Plotinus]
29. Religion / D. Religious Issues / 2. Immortality / b. Soul
Even the soul is secondary to the Intellectual-Principle [Nous], of which soul is an utterance [Plotinus]