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Haig–Simons income or Schanz–Haig–Simons income is an income measure used by public finance economists to analyze economic well-being which defines income as consumption plus change in net worth. [1] [2] It is represented by the mathematical formula: I = C + ΔNW. where C = consumption and ΔNW = change in net worth.
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: There is no non-trivial elementary embedding of the universe V into itself.
There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem, whose exact values are independent of ZFC. While nontrivial relations can be proved between them, most cardinal invariants can be any regular cardinal between ℵ 1 and 2 ℵ 0 .
Let's say the utility function is the Cobb-Douglas function (,) =, which has the Marshallian demand functions [2] (,) = (,) =,where is the consumer's income. The indirect utility function (,,) is found by replacing the quantities in the utility function with the demand functions thus:
He starts with a known initial wealth W 0 (which may include the present value of wage income). At time t he must choose what amount of his wealth to consume, c t , and what fraction of wealth to invest in a stock portfolio, π t (the remaining fraction 1 − π t being invested in the risk-free asset).
But [((A→(A→A))→(A→A))→A]→A is a tautology and thus a theorem due to the old axioms (using the completeness result above). Applying modus ponens, we get that A is a theorem of the extended system. Then all one has to do to prove any formula is to replace A by the desired formula throughout the proof of A.
This can be done in various ways, one example provided by Cantor's normal form theorem. Gentzen's proof is based on the following assumption: for any quantifier-free formula A(x), if there is an ordinal a< ε 0 for which A(a) is false, then there is a least such ordinal. Gentzen defines a notion of "reduction procedure" for proofs in Peano ...
Note in the later section “Maximum likelihood” we show that under the additional assumption that errors are distributed normally, the estimator ^ is proportional to a chi-squared distribution with n – p degrees of freedom, from which the formula for expected value would immediately follow. However the result we have shown in this section ...