Mathematical Logic - Science topic

If induction and deduction are mutually connected, what is the framework upon which they may be connected? Will such a framework of connecting induction and deduction be a mere perspective or something fundamental?

Courses in logic, philosophy of science, etc. begin always with a detailed study of induction and deduction. Most courses tell us that these are typically different kinds of reasoning. But are they?

I would happily obtain your arguments for or against the so-called mutual exclusiveness of induction and deduction.

I believe that relativising induction and deduction and connecting them with one another under some very general framework of thought will be a grand starting point to revolutionize the foundations of the whole of philosophy and science, and of logic and linguistic analytic philosophy in particular.

Please, check my P=NP proof for errors:

Please reply with a comment if you find any errors and if you find none, too.

The proof uses logic (incompleteness of ZFC), algorithms accepting algorithms as arguments, reducing SAT to another NP-problem, inversions of bijections.

The proof does not present a practically feasible NP-complete algorithm (so, I don't yet mine Bitcoin by it).

Please answer the question or recommend some articles.

Mathematical Logic is a key subject in many disciplines, and a good tool for the development of many mental function.

On the other hand, for many people it's hard to understand: is it the case of your students? Have you figured out why it is happening?

Stoic logic and in particular the work of Chrysippus (c. 279 – c. 206 BC) has only come down to us in fragments. To my knowledge the most accessible account is given in Sextus' Outlines of Pyrrhonism. Stoic logic certainly contained an axiomatic-deductive presentation of what we call today the 'propositional calculus'. The deductive system was based on both axioms and rules and appears to have been similar to Gentzen's sequent calculus. Certain accounts (by Cicero, if I am not mistaken) suggest that it included the analog of the 'cut rule'. There are tons of remaining questions. Was this propositional calculus classical or intuitionistic ? What type of negation did it employ ? Was it closer to relevance logics and many-valued logics or even to linear logic ? How did the Stoics treat modality ? What about the liar paradox ? How did they deal with quantification ? Was it in combinatory logic style or algebra of relations style ?

Is paraphrasing necessary in such cases, or is direct quotation with appropriate citation sufficient?

How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?

The idea is introducing student to the topic from a more general subject, and introduce the different structures of, say, Propositional Logic as a formal system, and from them deduce the Boolean Algebra (while introducing the interpretation of formulas), and further logical laws.

Prove or disprove that for every field F of characteristic two, there exist vector spaces U and V over F and mapping from U to V, which is F-homogeneous but not additive.

Will this be the final incarnation of this question?

My purpose in asking these questions is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it. And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?

Is a physical basis that necessarily requires constancy of the speed of light a logical impossibility, or is the constancy of the speed of light the result of ideas not yet found or applied?

Does isotropy require constancy of the speed of light?

Jensen’s inequality for concave and convex functions, implies for a logarithmic function maximal value when the base of the log is the system’s mean. Mathematically, this implies that the speed of light must be uniform in all directions to optimize distribution of energy. This idea has a flaw. Creation of the universe happened considerably before mathematics and before Jensen’s inequality in 1906. Invert the conceptual reference frame and suppose that Jensen’s inequality is mathematically provable in our universe because it is exactly the type of universe that makes Jensen’s inequality mathematically true in it. A mathematical argument based on Jensen’s inequality goes around in a circle. Are there reasons, leaving aside Jensen’s inequality (or even including Jensen’s inequality), that require constancy of the speed of light?

I need to look at the value of the string variable for each entity to determine which distribution to use to assign my next variable. I cannot figure out how to do this, I either get the error that Symbol name "if" is a reserved name, or I get "invalid expression the value entered must be in the form of a mathematical or logical expression. String values are entered by enclosing a sequence of characters in quotation marks."

Maybe I am using the wrong format, or I am not allowed to write such complex statements in the new value field of the assign block. I currently have:

if(Scale == "200pm")

{disc(.18848,5, .24315,2,1,1); }

else

if(Scale == "100nm")

{disc(.01347,5, .0756,4,.15225,2,1,1);}

else

{disc(.08429,5,.13,2,1,1)};

In my opinion dynamical system is part of prediction science , but i know prediction is only interested for logical mathematical reformulation , Now my question is : Since dynamical system investigate about futur results for any phenomena according to known data and prediction science is also seek about futur mathematical logic , then What is the difference between prediction science and dynamical system ?

Hello all,

I am aware of "formal methods" as it is used in computer science, to make sure our specifications are mathematically and logically sound (before we put them into a specific language). Isn't there a tool or form of notation so we can do that while we construct legitimate hypotheses for psychological research?

It is common knowledge that the construction of natural language questions (surveys about thinking and behavior), can be very questionable when it comes to construct validity. For example, did I cover everything? Am I even asking the right question? Trial and error without a formal test of our logic doesn't seem very efficient.

Does anyone know of a formal notation or research tool that lets us test natural language questions for how "sound" they are? After all, whether computer code (if/then we can do this) or human language (if/then we can assume that), it's pretty much the same logic. Thanks for any advice!

The well-known Zermelio's theorem states that every set can be well-ordered. Since arbitrary well-ordering is a linear ordering, from this theorem it follows the following corollary:

(A) An arbitrary set can be linearly ordered.

It is well-known that Zermelio's theorem is equivalent to the axiom of choice.

Question: Can Corollary (A) be proven without axiom of choice?

It is true that during the “one-to-one correspondence” operation between Real Number Set and Natural Number Set, after the elements in Natural Number Set have been finished up, the elements in Real Number Set are still a lot (infinite) remained:

 1, The elements in real number set are never-to-be-finished, endless, limitless------Real Number Set is really infinite!

2, The elements in natural number set are sure-to-be-finished, ended, limited-------- Natural Number Set is actually finite?!

There are still some other proofs of “one-to-one correspondence” operation between the two sets telling us a fact that the elements in many infinite sets are sure-to-be-finished, ended, limited and they are actually finite！

A typical tool and technique is Cantor's Power Set Theorem: all the elements in any infinite set can be prove “sure-to-be-finished, ended, limited and they are actually finite” in front of its own Power Set-------because during the “one-to-one correspondence” operation between the original set and its power set, after the elements in the original set have been finished up （finite）, the elements in its own Power Set are still a lot (infinite) remained!

Following eight unavoidable suspended conundrums in set theory have been troubling us human for a long time：

1.      Does the definition of each “infinite set” closely relate to “the elements’ nature, appearance and inter-relationship” ---------the very characteristics of the elements inside the very “infinite set”?

2.      If it does not at all and the elements in all different infinite sets are the same a heap of “infinite abstract stuff without any differences and relationship”, then how can we define and distinguish “different infinite set” and how can we believe that there may be quantity differences between “different infinite sets”?

3.      If it does and the elements in all different infinite sets are “the concrete carriers of ‘abstract infinite concept’ with differences and relationship” (unique characteristics), then how will these unique characteristics decide the quantity differences between “different infinite sets”?

4.      Are “infinite sets” in present set theory “actual infinite sets” or “potential infinite sets”?

5.      Are infinite elements in infinite sets “actual infinite many” or “potential infinite many”? If they are “actual infinite many”, how can we conduct the quantitative cognitions to them; and if they are “potential infinite many”, how can we conduct the quantitative cognitions to them?

6.      What kind of mathematical tool of “one-to-one correspondence” is? When we conduct the quantitative cognitions to different infinite sets with “one-to-one correspondence” tool, is it “one element corresponding to one element” or “many elements corresponding to one element” or “many elements corresponding to many elements”, is it “potential infinite many elements corresponding to potential infinite many elements” or “actual infinite many elements corresponding to actual infinite many elements” or “potential infinite many elements corresponding to actual infinite many elements”?

7.      How can we define “infinite” and “finite” if we agree with the idea and the operations in Cantor's proofs that many infinite sets in mathematics actually can be proved (turned into) finite set-------the elements in real number set are never-be-finished, endless, limitless and they are really infinite while those in natural number set are sure-be-finished, ended, limited and they are actually finite?

8.        What kind of mathematical tool of “limit theory” is? When we conduct the quantitative cognitions to different infinite sets, how can we use limit theory to analyze, manifest and treat those X--->0 elements’ number forms inside them (such as those X--->0 elements’ number forms in [0, 1] real number set)?

Paradox is a term that is used but not fully understood. A mathematical definition of paradox has been given by fuzzy logic, T= F= 1/2 but this appears to be incorrect.

I was looking for examples of first order sentences written in the language of fields, true in Q (field of rational numbers) and C (field of complex numbers) but false in R (field of real numbers). I found the following recipe to construct such sentences. Let a be a statement true in C but false in R and let b be a statement true in Q but false in R. Then the statement z = a \/ b is of course true in Q and C, but false in R. 

Using this method, I found the following z:=

(Ex x^2 = 2) ---> (Au Ev v^2 = u)

which formulated in english sounds as "If 2 has a square-root in the field, then all elements of the field have square roots in the field." Of course, in Q the premise is false, so the implication is true. In C both premise and conclusion are true, so the implication is true. In R, the premise is true and the conclusion false, so the implication is false. Bingo.

However, this example is just constructed and does not really contain too much mathematical enlightment. Do you know more interesting and more substantial (natural) examples? (from both logic and algebraic point of view)

The problem is explained in the attachment. Note that the Laplace distribution is a negative example to the problem, despite that its characteristic function is meromorphic in the whole  C-plane.

What I am curious to know is that, whether all the problems we have with arithmetics, like G2, still holds, if we change the concepts and properties of negation

Or

It can be wipe away by changing the definition of consistency.

A_Extended=[A 0 0 0 0 0;

I 0 0 0 0 0 ;

0 I 0 0 0 0;

0 0 I 0 0 0;

0 0 0 I 0 0;

0 0 0 0 I 0],

where I is identity matrix and 0 is zero matrix?

Assume that eigenvalues of A are known. 

If this is a difficult problem, can we say anything about being Hurwitz of A_Extended given that A is Schur?

I really appreciate your answers.

Best regards,

Mehran

I am looking for a function form with vertical asymptote at x=0, horizontal asymptote at y=y0>0 and an inflection point along with minimum in between. I know rational functions with third order polynomial both in numerator and denominator work. A sample graph is attached so you have an idea of the form I am looking for. The function form is (rational function)

y = y0*(1+a/x+b(1/x)^2+c(1/x)^3)

However, I am looking for a simpler function only with one or two adjustable parameters (I know y0 but a, b, and c are adjustable). I would like to have the function completely positive (y>0). The important thing is I would like to be able to find regions for adjustable parameters over which the function is strictly positive. Any comments or advice highly appreciated.

Thank you.

Theorems use to be falsifiable. For instance, the well-known Pythagorean one need not be true in non-euclidean geometries. Is there any scenario in which Gödel's theorems do not hold?

If they are true under all circumstances perhaps they are tautological statements. From tautologies we can only deduce tautologies, hence they are of no use.

It is analogous to Marcus equation- where L is lembda to calculate

∆G^‡=L/4 [1-(∆G^0)/L]^2

I am facing the problem in random case,as we know that eigen values of hermitian matrices should be real,and in my case when diagonal elements are random of hamiltonian,is it always necessary that deminant of Ham ,symmetric matrix is equal to product of eigen values or not.

Let me recall that, for a positive integer n, Cut(n) is the statement that, for each sequence of n-element sets, the union of all sets of this sequence is at most countable. Cut(fin) is the statement that countable unions of finite sets are at most countable. I am unable to deduce whether it is true in ZF that if Cut(n) holds for each positive integer n, then Cut(fin) also holds. Perhaps, there exists a mathematician who knows a model for ZF in which Cut(fin) fails and, simultaneously, Cut(n) holds for each positive integer n. I would be grateful for any helpful hint to give a satisfactory answer to my question. Regards, Eliza Wajch

The answer for the first order logic is in many elementary logic books. I looked in Wikipedia already. Anything that is somewhat related to this question would be appreciated.

If that will help here is more what I am doing. I look at satisfaction of second order formulas in models with infinite well ordered universe with pairing function and some additional model elements like predicates, constants and function symbols. 

In our present infinite related mathematics theory system, no “infinite things” can run away from “actual infinite” and “potential infinite”. So, are there “actual infinite many (big)” and “potential infinite many (big)” in our mathematics?

The exactly same question goes to infinitesimals: are there “actual infinite small (few)” and “potential infinite small (few)” in our mathematics?  --------- Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinitesimals” or “potential infinitesimals”?

The thousand—year old suspended infinite related paradoxes tell us: something must be wrong in present classical “actual infinite” and “potential infinite” related philosophy and mathematics!

Are Natural Mathematical Processes a necessary component in the running of the Physical Universe?  Could the Universe operate without them?  For example, are mathematical processes essential in the addition of distances or displacements?

This question is an attempt at rewording of my earlier question asking whether mathematics is intrinsic in Nature.  That ealier question was rather open to interpretational debate.  I am hoping, this time, I more acutely address the point I  intend.

At least from Zeno’s time, most people (except Zeno with his creative “actual infinite--potential infinite” related paradox families) have been ignoring a fact that our whole present infinite related mathematics theory system has been based on “actual infinite--potential infinite”. The indefinable and confused “actual infinite” and “potential infinite” have been troubling us human for at least 2500 years with all kinds of debates, magics, paradoxes,… ------ the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.

Take Un--->0 in Harmonic Series and dx --->0 in calculus for example: (1) are they same infinitesimals under the same infinite concept with the same infinite definition? (2) are they “actual infinite small (few)” and “potential infinite small (few)”?

We have “big” and “small” in our science and the related numerical cognizing way to universal things around us （one of the mathematical cognizing ways）. So, we have “very big (such as 1000¹⁰⁰⁰)” and “very small (such as 1/1000¹⁰⁰⁰⁾”, “extremely big” and “extremely small”… in mathematics. When “infinite” came into our science and mathematics, we naturally and logically have “infinite big (infinities)” and “infinite small (infinitesimals)”.

Many people think we really have had many different mathematical definitions (given by Cantor) for infinite: infinity (infinite big) in set theory. So, when talk about the mathematical definition for “infinite”, people only think about “infinite big” but negate “infinite small”.

Should the mathematical definition for “infinite” cover both “infinite big (infinities)” and “infinite small (infinitesimals)” or only for the half: infinite big (infinities)?

The example I have in mind is a pair of integrals, over an inseparable domain, that do not exist unless weighted by functions in Banach space. But even if they are not, they can inter-substitute to imply existence of the Fourier transform and its inverse.

I am looking  for recursive formula for special kind of operations on power series which is called the substitution of one series into another. The main idea is the following. We have two power series:

\sum_{k=1}^\infty{b_k y^k}=\sum_{k=1}^\infty{c_k  x^k}

and

y=\sum_{k=1}^\infty{a_k x^k}

I would like to obtain general formula for c_k using known coefficients a_k and b_k.

In the handbook I.S. Gradshtejn, I.M.Ryzhik "Tables of integrals, series and products", AcadPress, 2007, page 17, the formula (0.315) gives only the first four coefficients of the power series which is the substitution of one series into another.

I would be very grateful for any tips or links to papers which can give the answer for my question.

I would like to know a good text on non-standard models of Peano arithmetic. And also, any article about then. Thanks. 

Could contradiction play a role in quantum systems, as part of the mechanism of measurement, forcing a single random outcome from the spectrum of possibilities?

All ideas are welcome, including outrageous ones.

There have been two suspended questions challenging us human in present classical infinite related science system at least since Zeon’s time:

1, is it unavoidable that we mix (jump between) “potential infinite” and “actual infinite” whenever we cognize any infinite relating things?

2, do we treat potential infinite things or actual infinite things or jump between in mathematics quantitatively?

 Can we have “potential infinite sets”? If yes, can anyone give an example of “potential infinite set”?

In arithmetics or algebras that cannot be completed, if any statement is logically independent of the axioms, is it also mathematically undecidable.  Are these concepts identical?

The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.

Now, I've found two terms (not combinators) M and N with the following properties:

M = x

N = y

M can be obtained from N by replacing one or more occurrences of S with K

From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.

Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.

I know that there is a model M for ZF such that. for an uncountable set S in this model and for every collection $\{ (X_s, d_s): s\in S\}$ of metric spaces in this model, their product $\prod_{s\in S} X_s$ in M is metrizable in M. In particular, for an uncountable set S in M, the product $\mathbb{R}^S$ is metrizable, however, I have not found this result in the literature so far. I would be grateful if you could tell me whether you have located  it in the literature. If your answer is YES,  please, tell me where I can find this result. I know how to prove the result. 

If there is a game with no rules, does it need a rule saying it has no rules?  Or can it just simply have none.

If a game has 3 rules, does it need another saying it has exactly 3?  Or can it just simply have 3? 

For example, assume two entities P and Q, where we are using 'proof-by-contradiction' to validate P (by using Q).

Something like:

IMO, one can only use such scenario, when P and Q are independent of each other existentially.

In other words, can one use proof-by-contradiction in cases where Q's existence is dependent on P's validity?

For example, Q exists if and only if P exists.

In such case does the proof-by-contradiction still hold valid?

Does it help us understand concepts of randomness and probability by considering the possibility that randomness may be involved in certain strange systems, where only one outcome is possible.  The motivation for this question is that a system with 1 possible outcome is the limiting case of systems with  ....6, 5, 4, 3, 2, .. possible outcomes.  OR  If a system with only one outcome has no algorithm for that outcome, but that outcome has probability one, could randomness in such a system be meaningful?

If the decision between two choices is to be made, but neither choice is preferred over the other, because there is perfect symmetry between the two, then no information separates the choices and the only decision that can be made is a random one.

What kind of imperative can force such a decision, and does such a decision resist the imperative?

Quandles and racks are algebraic structures referred to in knot theory. Are there links to thorough introductions for researchers in how they are used in physics research?

PARITY is about whether a unary predicate of a structure has even numbers of elements in it. 

If a kind of logic can define PARITY, then there is a formula of this logic so that:

PARITY return True on a structure iff this structure is a model of this formula.

We have known that logics with counting can easily define PARITY.

But what about others without counting?

The following statement is well known in distributive lattices (D, 0, 1, ., +):

(1) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I.

Usually the proof of (1) follows by an application of Zorn Lemma. But then the proof yields a stronger version:

(2) Let F be a filter disjoint from an ideal I. Then there exists a prime filter F’ extending F and disjoint from I and for every x not in F’ there exists y in F’ such that x.y is in I.

I am interested if (1) and (2) are equivalent (or not ) in ZF. Any references?

Graph theory and mathematical logic, are both parts of Discrete Mathematics syllabus. Some logical equalities can be express by rooted trees.

I am looking for a proof such that:

Given a set of Horn clauses, show that there is a unit refutation from S if and only if S is unsatisfiable

Can The Physical Laws be regarded as a computational program that runs The Universe (Everything)?   If so, does the program need to generate The Physical Laws themselves?

This question is a followup on my earlier similar question referring to a computer simulation of The Universe. I give the link here below.

My purpose in asking these questions is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it. And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?

If a computer program were to be written to simulate The Universe (Everything), would the simulation need to include the computer and program?  And if so, what process could boot this Universe from a totally off state?

My purpose in asking this question is to motivate the kind of physical theory that accepts that Physical Laws are part of The Universe, as opposed to standing outside it.  And that rules governing The Universe must stem form The Universe itself. Otherwise, we should be asking: Where do the Physical Laws come from?  

I am studying the measurement problem in quantum mechanics.  I have a quantum theory based on a formal axiomatised arithmetic of scalars.  This is being considered under the disciplines of Mathematical Logic.  My arithmetic is a first-order theory and the tool I am using is first-order logic. I've had good success in finding logic in the quantum arithmetic, isomorphic with indeterminacy in quantum theory.  Specifically, I have identified, located and isolated logical independence in the wave packet for the free particle, just where indeterminacy shows up in experiments.  In order to make progress with collapse during measurement, I need to find mathematics that motivates a jump of the type, typified by the following.  Starting from the proposition: " there exists an x such that x=2 " -- jumping to the statement  " x=2 ".  Can anyone offer any ideas please.

In a formal arithmetical system, axiomatised under the field axioms, the square root of minus one is logically independent of axioms.  This is proved using Soundness and Completeness Theorems together. This arithmetic is incomplete and is therefore subject to Gödel's Incompletenss Theorems. But can it be said that the logical independence of the square root of minus one, is a consequence of incompleteness?

Nowadays, the consistency of theories is not demanded and in alternative we search for relative consistency. In the future things may change. In particular, in the answer 70 and more easily in answer 76, it was proved that set theory is consistent as a result of a relative consistency. There were published several datasets proving the consistency of set theory. In the last times it was publshed a paper in a journal, without success, since there is some inertia concerning the acceptance of the consistency of NFU set theory. It can be said that NFU set theory is consistent as the result of a relative consistency: since Peano arithmetic is consistent than NFU is consistent too. By a similar argument it can be prooved that set theory is consistent too: since NFU set theory is consistent then set theory is consistent. Thus, set theory is consistent, and since the related proof can be turned finite then we also prooved the Hilbert's Program, that was refered in many books on proof theory. There is an extension of set theory, the MK set theory, which is a joint foundation of set theory and category theory, two well known foundations of mathematics.  Once again a paper by myself with title "Conssitency of Set Theory" was rejected without a valid reason. This agrees with an answer given by me 26 days ago. With set theory consistent we can replace the use of models to prove the independence of axioms  (as did by Goedel and Cohen) by deduction in set theory.

I am a research scholar, working on transformer fault diagnosis and classification. Can anybody tell me how to calculate training time, in seconds, of transformers, using Back Propagation Neural Network, Fuzzy Logic and Adaptive Neuro-Fuzzy Inference System?

Like Aristotle's four causes, what are the different types of reasons?

I'm thinking there are one's based on sentiments, passionate and computational. That's as far as I've gotten. I'd love to hear comments additional ones. Also, here are just a couple stumbling blocks I've come across:

For example, what kind of reasoning did Gödel use to liberate himself from the very limitations which he argued were characteristic of sufficiently developed computational devices?

Or what about poor Gretchen in Goethe's Faust. She commits infanticide, yet she is the symbol of innocence throughout the epic. What kind of reasoning did she use to commit her action? Here we get into the very importance of law with the different types of reasoning.

Can anyone else elaborate or add to this small list of the different types of reasoning? 

It was true that mathematics was done in argumentation and discourse or rhetoric in ancient times. The 6 volumes of Euclid’s elements have no symbols in it to describe behaviors of properties at all except for the geometric objects. The symbols of arithmetic: =, +, -, X, ÷ were created in the 15th and 16th centuries which most people hard to believe it - you heard me write. The equality sign “=” and “+,-“ appeared in writing in 1575, the multiplication symbol “X “ was created in 1631, and the division sign “ ÷” was created in 1659. It will be to the contrary of the beliefs of most people as to how recent the creations of these symbols were.

It is because of lack of symbols that mathematics was not developed as fast as it has been after the times where symbols were introduced and representations, writing expressions and algebraic manipulations were made handy, enjoyable and easy.

These things made way to the progress of mathematics in to a galaxy – to become a galaxy of mathematics. What is your take on this issue and your expertise on the chronology of symbol creations and the advances mathematics made because of this?

http://Notation,%20notation,%20notation%20%20a%20brief%20history%20of%20mathematical%20symbols%20%20%20Joseph%20Mazur%20%20%20Science%20%20%20theguardian.com.htm

Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can't we define the domain X' = X u 2^X and for elements of x in X' define predicates:

SET(x)

ELEMENT(x)

BELONGS_TO(x, y) - undefined (or false) when ELEMENT(y)

etc.

Now, we can express sentences about subsets of X in the 1st-order logic!

Similarly we can define FUNCTION(x), etc. and... we can express all 2nd-order sentences in the 1st order logic!

I'm obviously overlooking something, but what actually? Where have I made a mistake?

If we have an n by n matrix called A. How do we know if there is an inverse matrix A^-1 such that the product A * A^-1 is the n by n identity matrix?

Please mention the name of some books and prerequisites?

Peirce’s law, (((P-->Q)-->P)-->P), is a law of the Pure Conditional Sentential Logic.

Q1: Who proved that it is not intuitionistically provable? Q1.1: What is the proof?

Q2: Who proved that it is not classically provable using only the standard -->Intelim rules? Q2.1: What is the proof?

Q3: What “meanings” have been suggested for it? How could it have been discovered? Why should anyone conjecture it?

Q4: What good is it? What results in logic use it?

Q5: Hassan Masoud observed that its unprovability using only the standard -->Intelim rules suggests that the standard -->Intelim rules should not be regarded as “determining the meaning" of the connective --> Has this or related observations been discussed in print?

It would appear that the probability of randomly selecting a specific element from a Transfinite Set would be zero. However, NOT making a selection at all, would also yield a zero probability. Thus the contradiction. Furthermore, this can be compounded further if we exercise the above consideration on countable and uncountable Transfinite Sets. Are some 'zeros' less than others?

ADDENDUM (Feb. 2016): Let us consider the above posed question as selection from a transfinite set of points on two Lines perpendicular or parallel:  L1 and L2 .  Now let us re-evaluate this in terms of Purdy's Conjecture, to wit.,  Suppose that n points are to be chosen on line L and another n points on line M. If L and M are perpendicular or parallel, then the points can be chosen so that the number of distinct distances determined is bounded by a constant multiply of n, but otherwise the number is much larger.

I would like to know what books are considered, from the personal experience point of view, more suitable for teaching Theorem Proving, when teaching this topic for Computer Science students. Usually it is a very complicated issue for students, specificaly those studing in first or second courses.

Recall that recursive definitions are those defining statements involving the defined object.

The defined object can occur either explicitly or implicitly in the recursive definition.

For instance, the definition.

x = positive number that coincides with x^2.

This recursive definition determines de number 1. The defined object x occurs implicitly. By contrast, in the following

definition

x = positive number that coincides with its powers.

contains x implicitly; because of, in this definition, the expression "its powers" is equivalent to x^1, x^2, x^3 ….

Another example is

x = set of all sets.

This definition is also recursive, because defines x as a set, therefore and the expression "all sets" involves x.

In fact, a recursive definition fits into the pattern

x = p(x)

where p(x) stands for any predicate involving x.

In general, recursive definitions are stated in several metamathematical topics without proving its consistence.

Nevertheless, if the pattern x = p(x) is regarded as an equation instead of a definition, then it is required

to show that such an equation has at least one solution. Why do not require also the existence proof in recursive definitions?

I'm currently writing a short essay on Formal Verification and Program Derivation, and I would like to include an example of an algorithm that seems to be correct under a good number of test cases, but fails for very specific ones. The idea is to present the importance of a logical, deductive approach to infer the properties of an algorithm, in order to avoid such mistakes.

I can't quite think of a good example, though, because I'd like something simple that a layman can understand, but all the examples I came up with are related to NP-complete graph theory problems (usually an algorithm that is correct for a subset of the inputs, but not to the general problem).

Could someone help me think of a simpler example?

For analysis purposes I need the coefficients of matrix powers C=A**m. Given a (n,n) matrix A with coefficients a(i,j) one can define recursive equations for the coefficients c(i,j):

For 1 <= i,j <= n and 1<=p<=m

c(p,i,j) = a(i,j) for p=1

c(p,i,j) = Sum(1<=k<=n, c(1,i,k) * c(p-1,k,j)) for p>1

but neither the sign of the result nor a closed formula can be seen immediately.

Does anyone have an idea to resolve the recursion?