From: JoeBiden.com
Date: Tue, Mar 3, 2020 at 11:38 PM
Subject: Bernie Sanders
To: ajay mishra
This week, Bernie Sanders’ campaign reported that they raised a staggering $46 million in February.



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From: JoeBiden.com
Date: Tue, Mar 3, 2020 at 11:38 PM
Subject: Bernie Sanders
To: ajay mishra
This week, Bernie Sanders’ campaign reported that they raised a staggering $46 million in February.



ᐧ
From: for Ajay’s eyes only
Date: Tue, Mar 3, 2020 at 11:01 PM
Subject: Super Tuesday: Official Texas Straw Poll
To: Ajay Mishra
We need [23] responses by 11:59pm.





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The first axiom states that the constant 0 is a natural number:
The next four axioms describe the equality relation. Since they are logically valid in firstorder logic with equality, they are not considered to be part of “the Peano axioms” in modern treatments.^{[5]}
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a singlevalued “successor” function S.
Peano’s original formulation of the axioms used 1 instead of 0 as the “first” natural number.^{[6]} This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.
The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.
then K contains every natural number.
The induction axiom is sometimes stated in the following form:
then φ(n) is true for every natural number n.
In Peano’s original formulation, the induction axiom is a secondorder axiom. It is now common to replace this secondorder principle with a weaker firstorder induction scheme. There are important differences between the secondorder and firstorder formulations, as discussed in the section § Models below.
The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in set theory or secondorder logic, and can be shown to be unique using the Peano axioms.
Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:
For example:
The structure (N, +) is a commutative monoid with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers.
Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:
It is easy to see that S(0) (or “1”, in the familiar language of decimal representation) is the multiplicative right identity:
To show that S(0) is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:
Therefore, by the induction axiom S(0) is the multiplicative left identity of all natural numbers. Moreover, it can be shown that multiplication distributes over addition:
Thus, (N, +, 0, ·, S(0)) is a commutative semiring.
The usual total order relation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:
This relation is stable under addition and multiplication: for {\displaystyle a,b,c\in \mathbf {N} }, if a ≤ b, then:
Thus, the structure (N, +, ·, 1, 0, ≤) is an ordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation “≤”:
This form of the induction axiom, called strong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals are wellordered—every nonempty subset of N has a least element—one can reason as follows. Let a nonempty X ⊆ N be given and assume X has no least element.
Thus, by the strong induction principle, for every n ∈ N, n ∉ X. Thus, X ∩ N = ∅, which contradicts X being a nonempty subset of N. Thus X has a least element.
All of the Peano axioms except the ninth axiom (the induction axiom) are statements in firstorder logic.^{[7]} The arithmetical operations of addition and multiplication and the order relation can also be defined using firstorder axioms. The axiom of induction is in secondorder, since it quantifies over predicates (equivalently, sets of natural numbers rather than natural numbers), but it can be transformed into a firstorder axiom schema of induction. Such a schema includes one axiom per predicate definable in the firstorder language of Peano arithmetic, making it weaker than the secondorder axiom.^{[8]} The reason that it is weaker is that the number of predicates in firstorder language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in firstorder language (in fact, most sets have this property).
Firstorder axiomatizations of Peano arithmetic have another technical limitation. In secondorder logic, it is possible to define the addition and multiplication operations from the successor operation, but this cannot be done in the more restrictive setting of firstorder logic. Therefore, the addition and multiplication operations are directly included in the signature of Peano arithmetic, and axioms are included that relate the three operations to each other.
The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms of Robinson arithmetic, is sufficient for this purpose:^{[9]}
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of a recursively enumerable set of axioms. For each formula φ(x, y_{1}, …, y_{k}) in the language of Peano arithmetic, the firstorder induction axiom for φ is the sentence
where {\displaystyle {\bar {y}}} is an abbreviation for y_{1},…,y_{k}. The firstorder induction schema includes every instance of the firstorder induction axiom, that is, it includes the induction axiom for every formula φ.
There are many different, but equivalent, axiomatizations of Peano arithmetic. While some axiomatizations, such as the one just described, use a signature that only has symbols for 0 and the successor, addition, and multiplications operations, other axiomatizations use the language of ordered semirings, including an additional order relation symbol. One such axiomatization begins with the following axioms that describe a discrete ordered semiring.^{[10]}
The theory defined by these axioms is known as PA^{−}; the theory PA is obtained by adding the firstorder induction schema. An important property of PA^{−} is that any structure {\displaystyle M} satisfying this theory has an initial segment (ordered by {\displaystyle \leq }) isomorphic to {\displaystyle \mathbf {N} }. Elements in that segment are called standard elements, while other elements are called nonstandard elements.
A model of the Peano axioms is a triple (N, 0, S), where N is a (necessarily infinite) set, 0 ∈ N and S: N → N satisfies the axioms above. Dedekind proved in his 1888 book, The Nature and Meaning of Numbers (German: Was sind und was sollen die Zahlen?, i.e., “What are the numbers and what are they good for?”) that any two models of the Peano axioms (including the secondorder induction axiom) are isomorphic. In particular, given two models (N_{A}, 0_{A}, S_{A}) and (N_{B}, 0_{B}, S_{B}) of the Peano axioms, there is a unique homomorphism f : N_{A} → N_{B} satisfying
and it is a bijection. This means that the secondorder Peano axioms are categorical. This is not the case with any firstorder reformulation of the Peano axioms, however.
The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF.^{[11]} The standard construction of the naturals, due to John von Neumann, starts from a definition of 0 as the empty set, ∅, and an operator s on sets defined as:
The set of natural numbers N is defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
and so on. The set N together with 0 and the successor function s : N → N satisfies the Peano axioms.
Peano arithmetic is equiconsistent with several weak systems of set theory.^{[12]} One such system is ZFC with the axiom of infinity replaced by its negation. Another such system consists of general set theory (extensionality, existence of the empty set, and the axiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
The Peano axioms can also be understood using category theory. Let C be a category with terminal object 1_{C}, and define the category of pointed unary systems, US_{1}(C) as follows:
Then C is said to satisfy the Dedekind–Peano axioms if US_{1}(C) has an initial object; this initial object is known as a natural number object in C. If (N, 0, S) is this initial object, and (X, 0_{X}, S_{X}) is any other object, then the unique map u : (N, 0, S) → (X, 0_{X}, S_{X}) is such that
This is precisely the recursive definition of 0_{X} and S_{X}.
Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called “nonstandard models“); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in firstorder logic.^{[13]} The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (secondorder) Peano axioms, which have only one model, up to isomorphism.^{[14]} This illustrates one way the firstorder system PA is weaker than the secondorder Peano axioms.
When interpreted as a proof within a firstorder set theory, such as ZFC, Dedekind’s categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any firstorder formalization of set theory.
It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative as Skolem in 1933 provided an explicit construction of such a nonstandard model. On the other hand, Tennenbaum’s theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable.^{[15]} This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possible order type of a countable nonstandard model. Letting ω be the order type of the natural numbers, ζ be the order type of the integers, and η be the order type of the rationals, the order type of any countable nonstandard model of PA is ω + ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
A cut in a nonstandard model M is a nonempty subset C of M so that C is downward closed (x < y and y ∈ C ⇒ x ∈ C) and C is closed under successor. A proper cut is a cut that is a proper subset of M. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.
When the Peano axioms were first proposed, Bertrand Russell and others agreed that these axioms implicitly defined what we mean by a “natural number”.^{[17]} Henri Poincaré was more cautious, saying they only defined natural numbers if they were consistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don’t define anything.^{[18]} In 1900, David Hilbert posed the problem of proving their consistency using only finitistic methods as the second of his twentythree problems.^{[19]} In 1931, Kurt Gödel proved his second incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself.^{[20]}
Although it is widely claimed that Gödel’s theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic using type theory.^{[21]} In 1936, Gerhard Gentzen gave a proof of the consistency of Peano’s axioms, using transfinite induction up to an ordinal called ε_{0}.^{[22]} Gentzen explained: “The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles”. Gentzen’s proof is arguably finitistic, since the transfinite ordinal ε_{0} can be encoded in terms of finite objects (for example, as a Turing machine describing a suitable order on the integers, or more abstractly as consisting of the finite trees, suitably linearly ordered). Whether or not Gentzen’s proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
The vast majority of contemporary mathematicians believe that Peano’s axioms are consistent, relying either on intuition or the acceptance of a consistency proof such as Gentzen’s proof. A small number of philosophers and mathematicians, some of whom also advocate ultrafinitism, reject Peano’s axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to be total. Curiously, there are selfverifying theories that are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all true {\displaystyle \Pi _{1}} theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the nonexistence of a Hilbertstyle proof of “0=1”).^{[23]}
From: Biden for President
Date: Tue, Mar 3, 2020 at 6:36 PM
Subject: a lot on the line today
To: ajay mishra
There are more than one thousand delegates being awarded across the country and we need to win as many as possible.



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REFERENCES
OK OLGA – I CANT EXPLAIN BEYOND – THIS TILL EVERN AFTER 2030 ET ALONE 2020 BECAUSE I WANT SOME PRIVACY