 Welcome back in the last lecture we discussed natural reduction method and then before that we discussed various other decision procedure methods such as semantic tableaux method etc. So using those particular kinds of methods we can discuss we can decide whether a given el form formula is valid or when two groups of statements are considered to be consistent etc. So in this lecture I will be focusing my attention on some of the important and celebrated results in the first order logic and these are usually there many theorems which are widely used in the first order logic the celebrated results in all one of the important results in this first order logic and in that context I will be dealing with three important theorems I am not going to so when you say that something is a theorem it has to find a proof in all I may not be able to produce proof of all the theorems that I am going to talk about in this course but basically I will be giving you some kind of idea about these three important theorems. So I will be talking about completeness theorem which is due to a famous logician could go there and then I will be talking about one of the important theorems which is known as compactness theorem and third one which is discussed widely in the literature of logic and no course of logic would be complete without this celebrated result so that is the Godel's incompleteness theorem. So Godel has been credited for both the completeness theorem as well as the negative result that is incompleteness theorem, incompleteness theorem is used widely used misuse and even sometimes even abused us. So we will be talking about so the basic ideas of these particular kinds of theorems in this lecture to start with before going any further we need to know something about these three important logical properties. So any logical formal system any logician logicians dream is to have these three particular kinds of properties. So in this this course is all about first order logic which includes predicate and prepositional logic the one of the important characteristics of first order logic is that first order logics are considered to be consistent they are considered to be sound and they are also considered to be complete when you come to second order logics etc and all is very difficult to establish the completeness property so what are these important logical properties we will just go through some kind of detail about these important logical properties all the time we are using these three properties when we are while we are discussing about semantic tableaux method or natural reduction method etc. So consistency the first property is consistency, consistency means that none of the theorems of your formal system contradict one another so that means you are not in a position to derive both p and not p and all if you derive both p and not p then your system is considered to be consistent so that means all true prepositions can be proven for example if some something is a true preposition or true formula that is tautology all tautologies are obviously valid formulas and all valid formulas have to we have to find a proof for all the valid formulas in that context we use natural reduction and some other semantic tableaux method etc we already done those things second important property is soundness the soundness property tells us that a given formal systems rules of proof will never allow us false kind of inference from the true premises if you start with the true premises you always end up with the true conclusion on there is no way you can generate a contradiction when you start with tautologies so that is the reason why we are cleverly chosen axioms into consideration and then these axioms are transformed by means of substitution rules etc and all modus ponens etc and then you it will become theorem all the theorems are also considered to be true each step of your proof is considered to be true the final step of your proof is also considered to be true soundness ensures that you start with the axioms which are obviously true tautologies you will never end up with contradictions suppose if you start with tautologies and end up with contradictions something wrong with your formal system there is something wrong with the proof or there is something which is mistaken and all in the proof mistakenly considered in the proof the third important property is completeness which means that there are no true sentences in the system that cannot or at least in principle be proved in the system suppose if you say that something is a true proposition it has to find a proof that means we are also ensuring that only true propositions will find a proof suppose if I give you a contradiction then ask you to prove and there is no way in which you can prove that particular kind of contradiction but if I give you a tautology like PRP PR not P etc obviously you can find a proof for that so now these are some of the important virtues every logicians would dream of but we will be seeing very soon that in the works of good Godel he showed that no useful system of arithmetic Godel was looking for this interesting properties in the context of arithmetic you could find an important result that is incompleteness theorem with which he could say that no useful system of arithmetic can be both consistent at the same time it can be complete now so it has to be inconsistent to be complete or it has to be incomplete to be consistent etc these interesting things that you will see it will be treated so now we will be focusing on three important theorems first is completeness theorem second is compact next theorem which has its consequence to another important logician lovenham Scholem lovenham's theorem and then we will end up with Godel's incompleteness theorem and its impact on in the area of philosophy so before going further this we need to know something about the first order logic so when we say first order logic it means this first order logic only allows quantified variables to refer to objects in the domain in the domain we have objects predicates functional symbols etc now so the quantified variables refer to only objects in the individual in the domain of discourse but not the predicates are functions if you also take into consideration predicates in functions etc you are talking about higher order logics second order third order etc first order logic is usually considered to be powerful and expressive system which has been used to formalize basic systems of mathematics arithmetic etc set theory and any other real closed field which are related to these two fields it is mostly suited for the mathematical reasoning you should not be under the impression that all kinds of reasoning can be captured within the scope of first order logic that is mistake so first order logic is not supposed to capture everything and all it basically as far as possible it tries to capture mathematical reason so now to start with we will begin with the compactness theorem compare this theorem tells us this thing suppose if assuming that S is considered to be set of well-formed formulas of first order logic we have seen what how we constitute well-formed formulas for the first order logic so now if you take S to be the well-formed formula then every finite subset of S if it is satisfiable then that particular kind of set S is also considered to be satisfiable so there are S is considered to be some kind of set of well-formed formulas and then let us assume that S1 S2 S3 are subsets of S if every finite subset of S that means S1 S2 S3 they are all satisfiable then obviously the conjunction S1 and S2 S3 is also considered to be satisfiable that is if a set of sentences is such that every finite subset of it has a model then the whole set is also having a model so that means you need not have to worry much about all the all the sets of formulas and all but you can even if you have finite subset of set which satisfies this particular kind of thing then you can extend it to the whole set you can say that whole set is having the model one lemma for this one compactness theorem is this thing if S is consistent set of sentences and obviously S has to have a model so proof of compactness theorem goes like this just I am giving you a general description of proof so you need to note that each and every theorem should have should find a proof and all although I am not producing proofs of all the theorems that I am trying to discuss my intention is to talk about only the general idea of these proofs where it is later we can see where it can be applied and all so proof of compactness theorem goes like this let us consider the contra position of compactness that means you take the negation of the statement that you have seen in the theorem of compactness that means if S is not satisfiable then there always exist some kind of finite subset of S3 S1 S2 or anything and that also should not be satisfiable and all if each and every finite subset is satisfiable then S is considered to be satisfiable suppose if one of the important use of this compactness theorem is that if you can check the satisfaction with respect to subsets of a given set that will serve our purpose so now let us assume that S is not satisfiable that means in the context of predicate logic we need to have a structure a such that S is not a logical consequence of a he does not a models does not model S so we thus merely need to prove that there exists a finite subset S prime which is a subset of S such that S S prime is not satisfiable so now we have a completeness lemma which says that if S is consistent then obviously S has to find a model so that we will make use of it here and we say that so if S has no model then obviously S has to be inconsistent if you do not find any model of that one obviously you know in all the interpretation that formula is false that is why it is considered to be inconsistent so for a theory to be inconsistent is just for it to prove that it leads to some kind of contradiction that means you have come up with the case where you have a formula and its negation as an outcome of your that particular kind of set of well formed formulas that means let let that particular kind of proof which consists of this contradiction be a sequence of steps which is represented by X so that X as this thing first step is X 1 X 2 X n and the n step is something like Phi and not Phi there is a contradiction that you got in your proof where n stands for n is a natural number that means in the finite steps this proof ends that means since X must be finitely long and only finite a finitely many excise can obviously appear in it so that means you came across some kind of contradiction that means you could have got it through finite number of steps in so such that excise some excise that you take into consideration which has that particular kind of contradiction that belongs to S now let S prime be a set of well formed formula S of S which appear in the proof that you are trying to talk about where you have a contradiction then S prime should also be finite like the one which we have we have seen in the case of S that means we also have Phi and not Phi which is which comes as an outcome of S prime that means S prime has to be inconsistent and thus the finite even the finite subset S is also considered to be not satisfiable so what the essential thing that you need to note is that if you want to show that S is not satisfiable you can show that you can take some finite subset of S and you can show that it is not satisfiable and that will serve over purpose in that sense we call formal system to be compact each and every finite subset of S has to be consistent satisfiable so that the whole set is going to be satisfiable wherever one particular kind of subset of S is not satisfiable and the whole set S is not going to be satisfiable what is the use of a comparison this theorem is considered to be an important rule especially in the context of model theory model theory is important branch in the area of mathematical logic it provides us with the useful method for constructing models of any set of sentences that is finitely consistent so it is one of the important applications of this compactness theorem. Now let us talk about another important and interesting result in the first order logic that is lowenheim's theorem suppose you take SS to be set of well-formed formulas that is considered to be satisfiable that means at least one interpretation that formula S is true then obviously if it is satisfiable then it has to find some kind of systematic table for X that S which obviously cannot close that means all the open branches are leading to satisfiability means if you have a formula in the predicate logic you constructed a tree diagram and you have open branches that means that open branch tells us that the given formula is satisfiable in that particular kind of interpretation that means S has to be satisfiable in a denumerable more domain so let this lowenheim's theorem tells us this thing if S is considered to be satisfiable that means it finds an open branch after using the tree rules then S is satisfiable in a denumerable domain that means you can number of steps would be finite in number or it is satisfiable in a denumerable domain and the corresponding theorem is this thing if all finite subset of S are considered to be satisfiable then the entire set is considered to be satisfiable in a denumerable domain. So there is another interesting corresponding theory theorem in the context of lowenheim it is called as lowenheim's upward lowenheim's column theorem so this tells us this thing if a theory S the formal system S has an infinite model then S this particular S has arbitrarily large kind of models if it is infinite model it will have it will take lot of time it will take infinite number of steps so S has infinitely large models for example if you want to see the one of the applications of this particular kind of a theorem it is like this formula for all x px for all x qx implies for all x px or qx is going to be tautology and hence it is valid formula but if you take the vice first of that one that is for all x px or qx implies individually for all x px or for all x qx that is considered to be an invalid kind of formula and if you want to establish that it is unsatisfiable and you if you construct a tree diagram in finite number of steps it will end and then you will have some kind of open branches that means if you deny the formula you will have the open branches this example we have taken into consideration in the last few lectures so we can easily construct from the open branch a counter example and if you see that counter example you will have you only require some kind of finite domain and all which involves one or two elements one or two parameters not a or b to establish this particular kind of thing so you will just require a finite domain to prove whether it is satisfiable or unsatisfiable one important corollary of this particular kind of theorem is this thing suppose if x is a logical consequence of set of well-formed formulas s then x is also considered to be logical consequence of some finite subset of s so you have some large set of set of formulas and then s1 consists of some kind of set of formulas which are subset of this particular kind of s then if x is a logical consequence of s then x is also logical consequence of some finite subset of s so how do we prove this particular kind of thing a proof goes like this suppose that x is a logical consequence of s that is what is given to us that means if you add not x to s that leads to unsatisfiable that means if something is this particular kind of thing s is x then if you add not x to it then this becomes unsatisfiable you started with the premises and you deny the conclusion obviously it leads to contradiction that means it is unsatisfiable so s union not of x obviously has to be unsatisfiable hence if it is unsatisfiable then if you constructed a tree diagram using table of proofs obviously tends in finite steps and all hence it will have a finite model s0 of s the capital s that is even s0 union not x is also considered to be unsatisfiable so therefore obviously x is also a logical consequence of s0 so if you want to show that s is considered to be unsatisfiable you can take subset of s and you can show that even from the subset of s also that x is has to be a logical consequence so this is what is compared next and lovin heim's theorem now let us come back to the celebrated result in the first order logic that is the Godel's incompleteness theorem before talking about Godel's incompleteness theorem let us talk about this particular kind of startling kind of passage which you will find it in one of the important and the path breaking papers by Godel it is on some kind of formally undecidable prepositions so here is an important question which is worth discussing so Godel says this particular kind of thing so he is of the view that he says the development of mathematics in the direction of greater precision has led to large areas of it being formalized so that proves can be carried out according to just few mechanical rules that means you started with few axioms that is what we have seen in the case of Russell white at axiomatic system or Hilbert Ackerman axiomatic system etc you started with some 4 or 5 kind of rules and then you have some substitution rules and some definitions that is schemas and then you generated number of theorems and all numerous theorems and all so which just simply requires few mechanical rules and all and then axioms of course to start with the most comprehensive formal systems to date according to Godel on the one hand are these things one is principia mathematical for whitehead and Russell and on the other hand we have Jeremy Lofrankel's system of axiomatic which we did not discuss in this course but the context of set theory is very important now these are the two examples that he takes into consideration now it talks about some of the important merits of these things of both these systems are so extensive that all methods of proof used in mathematics today can be formalized within these two frameworks that is it can be reduced to a few axioms and rules of inference the grand program is to reduce mathematics to logic and all the notions of mathematics if it is expressed in the language of logic that means you have axioms rules etc and all then you in a way you are reducing mathematics to logic in that sense arithmetic can be reduced to logic or geometry reduced to logic etc. So it would seem reasonable according to Godel that therefore to surmise that these axioms and rules of inference are sufficient to decide all mathematical questions that can be formulated in the system concern so he is praising this thing in this way but he goes on and says that in what follows is that it will it can be shown that obviously this is not the case this grand program might fail but rather that in both the cited systems there exist relatively simple problems of theory of ordinary whole numbers that cannot be decided on the basis of these axioms you will not be in a position to decide whether X is X can be proved or not X can be proved and all that leads to undecidability undecidability or incompleteness if all the truths are probable and all the proof all the things which can be proved are true then your system is considered to be complete but you can come up with simple examples where life is not that easy and all so you can come up with simple examples where at least Godel showed that the formal system is going to be incomplete. So now before talking about Godel's negative result that is incompleteness theorem let us talk about some of the important theorems due to Godel that is the completeness theorem so Godel's completeness theorem is considered to be the fundamental theorem in mathematical logic in fact he worked for his Ph.D. thesis this particular kind of problem completeness so completeness tells us that all the valid formulas are probable and all the probable formulas are considered to be true that is valid Godel's completeness theorem is a fundamental theorem in the mathematical logic logic it establishes a correspondence between semantic truth and the syntactic probability in the first order logic so on the one hand we have this particular kind of thing there are two symbols that we have used earlier suppose if you write like this X and Y this means Y is reduced from X that means X together with some kind of axioms etc well-formed formulas and then your you got Y what a finite number of steps in it is in that sense Y is reduced from X we are not trying to talk about what we mean by X etc so there is another symbol which is called as this thing this means X entails Y or Y is a logical consequence of X etc so that means under all interpretations in which X is true Y also has to be true there is no way in which X is true and Y is false enough if that is a case you write it like this the simple things which we have done in this particular kind of course so now the first one is trying to talk about probability and the second one the double tensile is referring to semantic truth so now Godel's completeness theorem states that a deductive system of first order predicate logic of course we did not discuss the axiomatic system in the context of first order logic but you can extend the prepositional logic with adding some more quantifies to it and you can come up with an axiomatic system that particular kind of a deductive system of first order predicate logic is considered to be complete in the sense that no additional inference rules are required to prove all logically valid formulas you start with only axioms and the transformation rules and the modus ponens etc and that is all you need to prove to check the validity of all the formulas now you do not require any other formula any other kind of any other kind of inference rule which is outside of these things it is in that sense within the system you can show that they are complete so now the converse of this completeness theorem is what is called as soundness so now soundness ensures the fact that only logically valid formulas are provable and only those which are considered to be truths are provable and those who are considered to be false of course obviously it cannot be proved in now within the deduction system it ensures that whatever you proved at the end of the day has to be will be true enough in all the true formulas are again provable enough so taken together these theorems imply that a formula is logically valid if and only if the conclusion of if and only if it is considered to be the conclusion of formal deduction with some kind of proof procedure methods that we have seen natural reduction method a semantic tableaux method we should be in a position to establish that you can find a proof for a given valid formula so now Godus completeness theorem tells us this thing it is considered to be the general version of completeness theorem for any first order theory T and any sentence is in the language of that particular kind of theory there is a formal deduction of S from T you can use my axioms and all these things if and only if S is satisfied by every model of T if it is the case then S can be deduced from T so now let us come back to the negative results that Godel has come up with so these are widely discussed in the literature of logic so they are the famous two income in his theorems the first one is first income in his theorem it tells us that it roughly states that any formal mathematical system which is sufficiently powerful to include for example arithmetic geometry etc contains a particular kind of statement which is true but it cannot be proved within the system that means it in company is results in all so that particular kind of formula is obviously true but it cannot be proved so this is Godel's completeness theorem links these two things to notion this is semantic notion and the other one is syntactic notion so something which is true has to be provable now Godel has come up with an interesting observation which is considered to be a path breaking result in the area of first order logic that is does all the true statements all the valid formulas find a proof that is the question that he asked and in the context of arithmetic especially there are some kind of obviously true prepositions we talk about some examples little bit later and they are obviously true our gut feeling says that they are obviously true but they are not provable that leads to incompleteness that means whatever all the valid formulas are considered to all the valid formula should find a proof if you are not able to prove it in the system is called as incomplete and the second incompleteness theorem is like this it states that such a system contains a proof of its own consistency suppose if you want to show that if you want to show the consistency of that particular kind of system and you can show it only when if and only if it is in fact consists inconsistent it has to be inconsistent to be complete so that seems to be a major blow to Russell Whitehead grand program of logicism and even Hilbert Ackermann's Hilbert Ackermann's axiomatic system Russell Whitehead is motivated by arithmetic as well as whereas Hilbert Ackermann is motivated by geometry so here incompleteness means that there are that there are in the language there are some undecidable kind of sentences that means sentences that could not be proved nor disproved in the theory there are undigisable kind of sentences that means there are obviously true statements in all but we are not able to prove we know that X is true in all but we are not able to prove X we are not able to prove X means we are not able to deduce X and in the same time we are not able to deduce even not X also at least one of these things should come as an outcome in your theory if both the things will come as an outcome obviously the system is considered to be inconsistent so that means you should be in a position to derive either X I should be in a position to derive not X if both are not derivable and all that leads to incompleteness now the second inclinement is theorem means that the theory makes it possible to deduce both a sentence that is alpha and its negation that is if you can deduce X and not X then that is contradiction is part and parcel of your system and that means you can derive anything in all so this is what we have seen earlier if 2 plus 2 is equal to 5 it is a humor we use this particular kind of example if 2 plus 2 is equal to 5 which is considered to be contradiction better Russell has showed that he is considered to be is a pope 2 plus 2 is equal to 5 better Russell is a pope so from contradiction anything you can derive so if your system has to be inconsistent to prove the completeness and there is something wrong in the system that leads to incompleteness theorem 2 that means anything if you can prove anything in all then the theory is going to be entirely useless Godel has taken into consideration some simple examples with which he showed that there are some obvious sentences which you come across within the formal system which is not considered to be either probable not probable enough so let us consider the sentence to the sentence can never be proved these are the sentences which he has taken into consideration and then he showed that you cannot deduce you and X that means you cannot prove X that means the sentence can never be proved this can be deduced or the sentence can never be proved that is it is not the case that the sentence can never be proved also will come as an outcome of your formal system that means both cannot be proved that leads to incompleteness now take up the first sentence you have no way of knowing whether in the context of nights and nails puzzles Smolian has nicely put this thing in the context of Godel's incompleteness theorem let us take this example you have no way of knowing whether I am a knight or a new based on this statement this statement is a statement that we have made that is you have no way of knowing whether I am a knight or a new based on this particular kind of statement so this how this leads to incompleteness theorem is the one which you are trying to see assuming that the statement is true that is you have no way of knowing whether you are a knight or name so then if the statement has to be true then the statement has to be made by the knight so what we are trying to do is we are discussing this Godel's incompleteness theorem in the context of nights nails puzzles nights always tells the truth's nails always tells lies it is never the case that night nails tells the truth's so it must be a knight who's saying it that's why this sentence is true that means if that sentence is true then this is what follows the logician won't be able to figure out that he is a knight because obviously the sentence is true so now once the logician realizes that he can't figure it out he will know that statement is true then now what happens here is that logician now figures out that it is night because the statement is true so that leads to a kind of paradoxical kind of situation therefore the statement cannot be true and must be a new it must be a new saying it because the statement is false the logician is able to prove that it's a new is able to prove both night and both nails and all so that leads to some kind of a problem related to incompleteness Godel also takes into consideration some simple examples such as let us if you consider a formal system yes that proves various English sentences but only true ones now let us take one simple example now we are asking this question what sentence that particular kind of thing be such that it has to be true but it is not probable in the system yes taken natural language sentence and all you're trying to come up with one particular kind of statement which is obviously we know that it is true but it is not probable in the system he takes this particular kind of example this sentence is not probable in the system that particular kind of thing is considered be a sentence which represented which is represented by yes now if the sentence were false that means it is false that sentence is not probable in the system then contrary to what it says what will happen is it would be probable in the system if the sentence is false and all and what it says is true it contradicts the given fact that the system proves only true sentences and all because here it is we are proving false sentences also and hence therefore the sentence cannot be false so you are assuming that sentence can be false it leads to a fact that it cannot be false because it's contradiction but suppose if you assume that it is true then what the sentence really says is this thing the sentence is not probable in the system that means there are some kind of sentences which are not probable in the system that means the sentence is true but not sentence is obviously true but it is not probable in the system so in both cases if you take this particular kind of sentence that this sentence is not probable in the system if it is false there is a problem if it is true then also there is a problem so the system has the sentence has to be true to be not probable and it has to be probable then it has to be false that means all false sentences you are able to prove and all the true sentences which cannot be cannot be proven and all so that goes against this particular kind of thing that all the valid formulas should find a proof all the true formulas should find a proof and all but you are not able to find a proof for this one because you are not able to find X you are not able to find not X also so in a different context you discussed this particular kind of result in the context of there are some important conjectures in mathematics which our gut feeling tells us that is obviously true but it has no proof you have to note that all the theorem should find a proof if you if you say that something is a theorem and it has no proof and that is not considered a theorem it is simply either it can be called as a conjecture or some kind of assumption or your belief or etc so in mathematics proof means it has to sorry theorem means it has to have a proof. Now Goldberg's conjecture is like this every even number greater than 4 can be written as some of two odd prime numbers everyone knows even the elementary high school going student also knows that this seems to be the case and obviously the case that means if you take six into consideration that is nothing but some of two odd numbers that is three plus where three is considered to be odd odd prime number on the same way 14 if you take into consideration it is 7 plus 7 so we know that every even number can be expressed in terms of greater than 4 can be expressed as some of two odd prime numbers the prime numbers obviously odd only so now every number greater than 2 this is also another way of putting it is a Goldberg conjecture every number greater than 2 can be written as some of three prime numbers so now in the first case 8 for example even number can be expressed as 3 plus 5 where 3 is odd and 5 is odd prime numbers in 20 for example 13 plus 7 or it can be written as 17 plus 3 as well in the second class even number for example 42 is represented as 23 plus 19 etc. So our gut feeling tells us that this is obviously the case whatever number that you are taking into consideration even number that can be expressed in terms of some of odd prime numbers but till to date we do not have any proof for this particular kind of obviously known fact and all is considered to be true but we do not have any proof so we have some kind of reference which we will get it in the letter by Euler to Goldberg he says this there is absolutely little doubt about this particular kind of result that it is true and all there is no doubt that in a gut feeling says that that is obviously true and all because whatever number that you have even number that you have taken into consideration it is it is expressed as some of two odd prime numbers in all but what is the guarantee that there exists some kind of big number where it cannot be expressed as some of odd prime numbers in all we do not know at this moment so there is it is not proved it so there is little doubt that this result is true this is what Euler is expressing to Goldberg that every even number is some of two prime numbers I consider this as entirely certain theorem in spite of that I am not able to demonstrate that this is considered to be a theorem that is not able to prove either X you are not able to prove not exercise so this these are some of the things which poses some kind of challenges are set some kind of limits to the first order logic where we have said that all the true propositions are considered to be proved we will end this lecture with by saying some kind of insights from this Godel's in company so you need to note that we are not it proved this Godel's in company in a serum and all it requires entirely a different kind of course to talk about the celebrated result of good Godel but I only given the general idea of this Godel's in company in this theorem some of the insights we can talk of and then we can end this lecture so Godel's income in a serum establishes a kind of first of all occurs the company in a serum which is mostly ignored by many logicians in all is also considered to be one of the celebrated results and all mostly people are interested in the negative result in company next etc. But completeness also you worked out in in his PhD thesis it establishes a correspondence between semantic truth and the syntactic probability single turnstile and double turnstile in the first order logic whatever is probable is true whatever is true is probable and we have also seen that completeness theorem is applicable only for first order logic in the case of if you talk about variables ranging over predicates function symbols etc. We are talking about second order logics it loses some of the interesting features like completeness consistency etc. It is very difficult to establish these important logical properties the theorems become incomplete in the case of higher order logics third order logic second order logic etc. So now Godel's incompleteness theorem also says that a consistent system can never be complete if it has to be complete it has to be inconsistent that means consistency and completeness can never go together consistency means you are not in a position to derive both XN if you are able to derive both XN nautics the system is called as inconsistent either you have to derive X or you have to derive nautics completeness tells us that if something is true it has to find a proof this both the consistency and completeness never go together it's giving us some kind of impression that they don't go together. So now if you observe if you have if you have some kind of closer scrutiny over our own individual lives and all we realize that the realities of our life with its promises and uncertainties etc all can be expressed only in higher order higher in order so that means there is some kind of incompatibility between consistency and completeness so all these things there is some gap between consistency and completeness so now the real question comes to us is which will be rising in a while from now so now the mathematical notions of completeness and incompleteness together provides us insights for science and religion dialogue for a better comprehension of reality in life obviously we see that there is some kind of we know that consistency and completeness never go together with as a result of this Godel's incompleteness theorem a few fit pertinent questions one need to ask oneself before we end this lecture we have to ask this particular kind of questions in the context of Godel's incompleteness theorem can our life be consistent and complete at the same time taking into consideration higher all complexities etc and all the second question that we can ask is should we try for consistency and completeness together our third question is what is the rule of the rule of the experience of inadequacies inadequacies and incompleteness in making one's life more meaningful and enriching officially our goal is to make our life meaningful and enriching but we have a celebrated result where consistency and completeness cannot go together so in what sense we can make our lives meaningful and enriching so we will end this lecture with a summary of what we began with some of the important theorems of first order logic some of the important theorems we are not trying to cover all the theorems we started with compactness theorem and then we moved on to completeness theorem and then we discussed the celebrated negative result by Godel which sets limit to the grand program of logicism that means all the mathematics can be reduced to logic so we discussed Godel's incompleteness theorem and then to certain extent we discussed some of the philosophical impact of Godel's incompleteness theorem so the questions that I rise pertinent questions that requires valid answers it requires satisfactory answers so this lecture is not going not talking about answers for these particular kinds of questions