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I guess what you are saying is that there exist a set of mathematical theories of comparable power to ZFC that all agree on the truth value for any finite arithmetical statement and this is a form of absolute mathematical truth. Did I get that right? Any theory which did not agree simply would not belong to this set.

Apparently, Scott does not consider the possibility that there might be two natural extensions of ZFC, that have contradicting arithmetical theorems. But, it is for instance known that Reinhardt cardinals are consistent with ZF but not with the axiom of choice, on the other hand we have measurable and supercompact cardinals etc. So, such a possibility cannot be excluded in principle, although it is not clear if there are arithmetical discrepancies between the currently used theories.

Lucent Primary Arithmetic Book Pdf Download

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This is a rather elementary paper, and does not consider large cardinal constructions so these models of set theory might not be "natural", but it illustrates the point that (even when you have 1 agreement), higher order arithmetical statements are not determined. The two set theories (or the parts used for proofs) might both seem natural by different kind of transfinite intuitions, in principle (this is not a mathematical statement).

Would you be willing to throw out ZFC and its implications for transfinite numbers and Banach-Tarski for a formal system that similarly upheld your intuition for arithmetical soundness, but did not allow such transfinite silliness? Even eager?

In the end of the paper, there is nice overview about philosophical positions held by various mathematicians/logicians. From strict Platonists, to middle of the road Feferman, to Hamkins who rejects arithmetic determinancy to ultrafinitists (relevant in the case the world is finite). But the point is, these are PHILOSOPHICAL positions.

Mathematics can only suggest (even prove as in this paper) that one kind of argument sometimes can not be extended beyond its most narrow scope, as you did in #46 if you wanted to use intuition about Turing machine halting to ALL first order arithmetic and arithmetic soundness and determinancy.

The more explicit of the two given constructions/proofs of Theorem 1, due to Woodin, is on the page 11. I doubt you could say that the formula they construct is universal-existential, as only its Godel number is constructed and it is SOME formula of first order language of arithmetic. So in proof of Theorem 1, there is no explicit expression for formula in that paper, and it is certainly not said to be universal existential, but it is of first order number theoretic and cannot be purely universal or purely existential by its properties. The construction is indirect.

Personally, I am not convinced of the absolute truth of arithmetic statements. But I know that it would be hard to convince Scott of my position, and Scott probably knows that it would be hard to convince me of his position. I think:

In a certain sense, nave set theory is inconsistent, but nave arithmetic is not. Hence independence of set theoretic statements from set theory is seen as a sort of fault of the statement itself, while independence of arithmetic statements from a given formal system is seen as a fault of the formal system.

So, we can extend this logic further up the hierarchy, to conclude that if natural numbers are the same in the two models AND are standard natural numbers, then the first order formulas of arithmetic hold the same in the two models.

Suppose we have an extension of ZFC (it will be non recursive) as a formal system, that has ALL purely universal true (in ordinary sense) formulas about arithmetic as axioms. Of course, it will also prove all purely existential such formulas that are true. Lets call this theory M. It might not be complete, but it is complete as far as these lowest level quantified formulas of arithmetic go.

Tarski elementary geometry is math w/o arithmetic. So PA not required for proper math. Could presumably extended to yield PA, but, more presumably, be extended to yield NON-PA? Thus would have proper math where 2+3 != 5. Arithmetic is boring imho ?

Why should the possibility of a formal system which is not sound (indeed, so unsound as to be able to prove 2+3!=5) make anyone doubt the absolute truth that two plus three equals five? Any more than the fact that sometimes human beings make mistakes when they do arithmetic?

Also, given the above, who is to say that we have access to all arithmetic computations in our universe? What if there are computations involving the standard integers that are foreclosed by the program running this universe? We would be completely cut-off from them even if our intuition insists that it is complete when it comes to the standard integers?

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements.[8] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra[a] and infinitesimal calculus were introduced as new fields. Since then, the interaction between mathematical innovations and scientific discoveries has led to a correlated increase in the development of both.[9] At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method,[10] which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes.[18] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics.[19]

At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics.[23][10] The 2020 Mathematics Subject Classification contains no less than sixty-three first-level areas.[24] Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations.[25]

Number theory began with the manipulation of numbers, that is, natural numbers ( N ) , {\displaystyle (\mathbb {N} ),} and later expanded to integers ( Z ) {\displaystyle (\mathbb {Z} )} and rational numbers ( Q ) . {\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but nowadays this term is mostly used for numerical calculations.[26] Number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid of ancient Greece and Diophantus of Alexandria.[27] The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.[28]

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid.[42] The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether.[43] (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)

This became the foundational crisis of mathematics.[57] It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have.[23] For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning.[58] This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910.[59]

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[74] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time.[75] 17dc91bb1f