"The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency." - Wikipedia
In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that is true of arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this using a process known as Gödel numbering. Wikipedia
"Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and utilising data and information that are vague and lack certainty." - Wikipedia
We might also think of incompleteness systems as asymmetrical to their intended design for symmetry. Similar to the bow tie which accompanies a tuxedo, bow ties are not meant to be perfectly straight but a bit off, a bit imperfect. Thus I had mentioned Einstein's Cosmological Lamba Constant. He purposely introduced it to account for the universe's vacuum energy densities and gravitation push-pulls on itself. The universe isn't perfectly in equilibrium throughout it's vastness. It's off a little bit here-and-there. It holds some asymmetry within it. To account for its imbalance Einstein pushed a variable into his relativity formula to help "balance" out its messiness so that it might become "perfect". In doing so he believed what he was doing was correct but stated later even he made certain assumptions, and held expectations, which do not conform to perceived truth. Science is still trying to work this out having now understood it was in Einstein's assumptions that he erred.
"If a truth theorem in complete, it's closed. If a truth theorem is incomplete, then it's open." - re slater
Hermeneutics of faith, the counterpart to hermeneutics of suspicion, is a manner in which a text may be read. It was the traditional or predominant way of reading the Bible for at least the first fifteen hundred years of Christian history. Both interpretive approaches combined are necessary for a complete knowledge of an object.Hans-Georg Gadamer, in his 1960 magnum opus Truth and Method (Wahrheit und Methode), offers perhaps the most systematic survey of hermeneutics in the 20th century, its title referring to his dialogue between claims of "truth" on the one hand and processes of "method" on the other—in brief, the hermeneutics of faith versus the hermeneutics of suspicion. Gadamer suggests that, ultimately, in our reading we must decide between one or the other. [re slater - or to both equally in tension...]According to Ruthellen Josselson, "(Paul) Ricœur distinguishes between two forms of hermeneutics: a hermeneutics of faith, which aims to restore meaning to a text, and a hermeneutics of suspicion, which attempts to decode meanings that are disguised." Rita Felski posits that Ricœur's hermeneutics of faith did not become fashionable because it appeared dismissive of the work of critique that defined an ascendant post-structuralism.In his early essay "The Universality of the Hermeneutical Problem" and especially his Wahrheit und Methode (Truth and Method), conservative German philosopher Hans-Georg Gadamer asserts that one is always deciding between a hermeneutics of faith (truth) or a hermeneutics of suspicion (method) when engaged in the act of reading.
- Allegorical interpretation of the Bible
- Anagoge
- Asian-American biblical hermeneutics
- Christian apologetics
- Biblical accommodation
- Biblical law in Christianity
- Biblical literalism
- Biblical studies
- Brevitas et facilitas
- Formulary controversy concerning Jansenius' Augustinus in the 17th century
- Jewish commentaries on the Bible
- Literary criticism
- Literary theory
- Narrative criticism
- Patternism
- Postmodern Christianity
- Principles of interpretation
- Quranic hermeneutics
- Summary of Christian eschatological differences
- Syncretism
- Trajectory Hermeneutics
Set Symbols
A set is a collection of things, usually numbers. We can list each element (or "member") of a set inside curly brackets like this:
Common Symbols Used in Set Theory
Symbols save time and space when writing. Here are the most common set symbols
In the examples C = {1, 2, 3, 4} and D = {3, 4, 5}
Symbol | Meaning | Example |
---|---|---|
{ } | Set: a collection of elements | {1, 2, 3, 4} |
A ∪ B | Union: in A or B (or both) | C ∪ D = {1, 2, 3, 4, 5} |
A ∩ B | Intersection: in both A and B | C ∩ D = {3, 4} |
A ⊆ B | Subset: every element of A is in B. | {3, 4, 5} ⊆ D |
A ⊂ B | Proper Subset: every element of A is in B, but B has more elements. | {3, 5} ⊂ D |
A ⊄ B | Not a Subset: A is not a subset of B | {1, 6} ⊄ C |
A ⊇ B | Superset: A has same elements as B, or more | {1, 2, 3} ⊇ {1, 2, 3} |
A ⊃ B | Proper Superset: A has B's elements and more | {1, 2, 3, 4} ⊃ {1, 2, 3} |
A ⊅ B | Not a Superset: A is not a superset of B | {1, 2, 6} ⊅ {1, 9} |
Ac | Complement: elements not in A | Dc = {1, 2, 6, 7} When = {1, 2, 3, 4, 5, 6, 7} |
A − B | Difference: in A but not in B | {1, 2, 3, 4} − {3, 4} = {1, 2} |
a ∈ A | Element of: a is in A | 3 ∈ {1, 2, 3, 4} |
b ∉ A | Not element of: b is not in A | 6 ∉ {1, 2, 3, 4} |
∅ | Empty set = {} | {1, 2} ∩ {3, 4} = Ø |
Universal Set: set of all possible values (in the area of interest) | ||
P(A) | Power Set: all subsets of A | P({1, 2}) = { {}, {1}, {2}, {1, 2} } |
A = B | Equality: both sets have the same members | {3, 4, 5} = {5, 3, 4} |
A×B | Cartesian Product (set of ordered pairs from A and B) | {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)} |
|A| | Cardinality: the number of elements of set A | |{3, 4}| = 2 |
| | Such that | { n | n > 0 } = {1, 2, 3,...} |
: | Such that | { n : n > 0 } = {1, 2, 3,...} |
∀ | For All | ∀x>1, x2>x |
∃ | There Exists | ∃ x | x2>x |
∴ | Therefore | a=b ∴ b=a |
Natural Numbers | {1, 2, 3,...} or {0, 1, 2, 3,...} | |
Integers | {..., −3, −2, −1, 0, 1, 2, 3, ...} | |
Rational Numbers | ||
Algebraic Numbers | ||
Real Numbers | ||
Imaginary Numbers | 3i | |
Complex Numbers | 2 + 5i |
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