By: Patrick Grim read in 2016

1 Plantinga's and Guanilo's arguments for and against Anselm. Greatest possible island and greatest possible natural number. Augments Practical Number Theory.

2 Opening lines of Wittgenstein's *Tractatus*

5 "[T]hese arguments *suggest* ... that notions of a totality of truth or of total knowledge must prove incoherent." Just as I thought

12 The technical error explained in footnote 14 does not (IMHO) simplify things but instead makes the argument hard to follow.

15 "What if we add an *infinite* number of additional values? Will *that* offer an escape?" The "infinite valued logic" of Rescher seems useful to me. It is similar to my scale of belief expressed in My Beliefs About God. To be useful, however, it needn't be "infinite" valued; any practical large number will suffice as long as it has enough resolution to quantify your data. If you can estimate the probability (or truth value) to within 1%, then 100-valued logic would be adequate. Infinity brings on its own problems

17 "[T]he greater the '*n*' of an *n*-valued logic, the greater power of expression is demanded in its Strengthened Liar." Yes. With absurdity as its limit.

18 "[T]he appeal to many-valued logics will prove inadequate as a reply to Liar-like arguments regarding possible worlds, omniscience, and a totality of truths." I agree. The notion of infinity (perfection, omniscience, completeness, ...) is the real culprit. Rescher's notion still seems to me as being useful in forming a new definition of 'belief'.

20 "A propositional account alone offers no theory of *why* certain sentences will supposedly fail to express propositions, and in that sense, it seems too *thin* to qualify as a satisfactory solution." I offer such a theory in my On A Deep Philosophical Question. It is based on a distinction between sentences and propositions. The former is physical and the latter mental. It requires Cartesian Dualism.

21 "Here, however, I want to press a further difficulty--" The solution in my essay handles this difficulty too.

22 "The propositionalist defender of omniscience ..." But not all propositionalists defend omniscience.

22 "We can also make the propositionalist's predicament still more telling." This, too, is defeated by my incompleteness argument.

23 "One consequence of a propositional account is thus that the propositional status of sentences—whether a sentence does or does not express a proposition—will often prove inscrutable." I agree. But missing here is the recognition of a prior condition, which must be established before we can even ask whether or not a sentence expresses a proposition. That condition is that all terms in the sentence be well-defined. In these self-reference examples, they cannot be.

25 It seems to me that all the difficulties discussed in Section 6 can be handled by accepting incompleteness while maintaining consistency.

25 "Graham Priest, for example, suggests that "trying to solve the paradoxes may be the wrong thing to do."" I strongly disagree.

27 "As advocates of inconsistency-tolerant systems have standardly argued, we *may* be forced to accept contradiction when there is no alternative. But in the Liar-like arguments here at issue, there clearly *is* an alternative: we can abandon notions of a set of all truths, of set-theoretic possible worlds, and of omniscience." And, I think that can be accomplished simply by abandoning the notion of infinity.

31 "In a number of important respects, the Tarskian hierarchy is an improvement over Russell's theory of types."

31 "But a Tarskian hierarchy still does not come without significant cost. It is technically limited to finite levels, those levels are fixed intrinsically and in advance rather than floating on wayward facts as forms of the Liar often do, and it is unable to deal intuitively with cases in which Richard Nixon and John Dean call each other liars." I think all those objections should be acceptable. We should admit that everything is finite and that some truths will never (at our level) be known.

32 "[A]ppeal to hierarchy appears ad hoc. Familiar notions of truth, belief, membership, and the like simply don't seem to come with anything like subscripts attached" Traditional belief systems come with a two-level hierarchy of celestial/worldly and now science, although denying that, has introduced emergence, chaos, and the quantum/classical distinction. These should all be taken as clues that maybe a Tarskian hierarchy obtains after all. I believe it does.

32 "Second and for my purposes more important, a Tarskian hierarchy effectively prohibits any global notion of truth or knowledge." That is an acceptable loss. Let's just live with it.

32 "Were we to adopt a similar hierarchical treatment for levels of divine knowledge or for truth in belief contexts, neither the relevant predicate of any true (41) nor of the claim that God believes that (41) is true could be assigned any level.

God believes all truths. (41)"

No problem IMHO we are well past the time when we should deny omniscience.

34 "A nice technical aspect of the Kripkean approach is that after sufficiently many stages, indeed, transfinitely many, the process saturates at a fixed point; the sets of true and false sentences are the same as at the preceding level." This is an unwarranted, and IMHO, mistaken assumption. Transfinite numbers cannot be consistently defined, as Cantor, Russell, and Gödel discovered.

37 "As is clear in Russell, Tarski, and Kripke, however, the hierarchical approach leaves no place for a set Ѳ of all truths, an actual world corresponding to a maximal truth set *A*, or a notion of omniscience." A perfectly reasonable and believable conclusion.

37 "Both of Barwise and Etchemendy's models rely on Aczel set theory—ZFC/AFA, or Zermelo-Fraenkel set theory with an antifoundational axiom--" Does the theory really include AC? Either way, what is the AFA?

39 I don't understand how he can claim that semantic closure implies inconsistency, but it rings true and I believe it.

46 "In the end, an appeal to hierarchy of one sort or another seems to emerge as the strongest candidate for an adequate treatment of the Liar." Yes. We should pursue the implications.

46 "Such an approach, however, far from saving a totality of truths, set-theoretic worlds, or omniscience, *itself* excludes these as incoherent." Yes. Those are among the first such implications we should take seriously.

49 "Let us start by borrowing the syntax (alphabet and grammar) of any system adequate for arithmetic." What does "adequate for arithmetic" imply? Does it imply the Gödelian condition of the existence of an infinite set of integers?

50 "Ɐ*x*[*x* ≠ 0 → Ǝ*y*(*x* = *Sy*)]" Just as I thought. This 3rd axiom implies an infinite set which in turn introduces inconsistency into the system, IMHO, destroying the value of the discussion. What needs to be defined is the notion of 'Ɐ'.

50 "On the standard interpretation we take all of these to be true, of course, both because they seem transparently true (in the vernacular, the second amounts to, 'Zero is the successor of no natural number', for example) and because we take *arithmetic* to be true." I do not.

50 "Note now two key facts. The first is this. All strings of symbols..." It's a stretch to claim this is a "fact" when you have not defined 'all'.

50 "Assuming a particular Gödel numbering, ..." This exposes Achilles' heel.

51 "The constructed list, composed merely of axioms borrowed from *Q* and the claims indicated in the three schemata above, is provably inconsistent." Clever use of the adverb 'merely'. It draws attention away from that Trojan Horse of the third axiom.

53 "Arithmetic, surely, is not to be abandoned." And why not? My Practical Number System could replace it.

54 "with '*Kap*' read as '*a* knows that *p*'." I think the footnote to this line raises an important point: facts cannot be known, or even make sense, in isolation. They can only do so in some context.

56 "But here again it appears that these cannot be consistent with axioms for arithmetic." And here again I suggest replacing Arithmetic with Practical Numbers.

57 "Here, however, I want to press a tidier objection against any propositional reply to the Knower. It won't work." Ha! Ha! Now that __is__ tidy.

59 "It has been suggested, for example, that a broad range of terms call for hierarchical treatment, including perhaps truth, necessity, knowledge, belief, and other propositional attitudes." Yes. It has been suggested by me. Remember the levels! Hierarchies of dimensions, manifolds, and Rosenberg's Natural Individuals.

60 "A propositional approach *can* escape the Strengthened Knower, then, by means of auxiliary recourse to hierarchy." And I propose Rosenberg's hierarchy for knowers, a multi-dimensional hierarchy of manifolds for space-time, and a combination for the Cartesian Dual world.

60 "But a propositional treatment tempered with hierarchy will also inherit the difficulties of a purely hierarchical response." The problem only occurs if you admit infinity. If everything in both hierarchies is kept finite, then we can get consistency at the expense of completeness. This is consistent with *The Incomplete Universe*.

64 "We *could* legitimately treat Burge's example of a global principle of bivalence as a genuine statement, of course, were we to treat it in terms of universal quantification over indexical subscripts." The problem as I see it, once again, is the tacit assumption that there exists an infinite set of subscripts. Burge depends on that assumption by mentioning *i*+1.

64 "Schemata in fact *are* often interpreted as involving implicit quantification in a metalanguage, and it is far from clear how *else* to interpret them." It's not the interpretation that is the problem but the quantification. If we assume only a finite set of potential indices, we maintain consistency at the expense of completeness.

65 "But if we take '*i* + 1' at face value, this appears to be simply a further schema, ..." Yes, but more fundamentally it tacitly implies an infinite set of integers.

66 "The first of our options here, the "true" option, licenses a list that contradicts that of (33) itself at an infinite number of points." As I was saying.

67 "Despite recourse to schemata, we are left with all the straightforward consequences of a simple hierarchy." Yes! And the most straightforward such consequence is the existence of a highest level and of a finite number of levels.

73 "Let us also assume *G* to be standard in other regards: to contain a merely denumerable alphabet, for example, only finite formulas, and rules of inference restricted to finite sets of premises." The adverb "merely" disguises the seriousness of assuming denumerability.

73 "[W]e might insist that the axioms of *G* be finite. Instead, however, let us impose the weaker requirement that the axioms of *G* be recursively enumerable." That's a shame IMHO. It seems that the finite case would be more fruitful.

78 "R. G. Jeroslow's "experimental logics" ... transcend the limits of standard systems in being interpretable in effect as dynamic rather than static, as progressively building over time. As such a logic develops, axioms and even rules of inference can be withdrawn or supplemented. Thus, in a sense experimental logics offer models not merely of ideal knowers but also of ideal learners." This sounds very much like how I think reality actually works. And yes, I think trial and error processes are in play as Grim mentions in footnote 21.

78 "Such logics don't seem to offer any bright hope for modeling a developmental march toward a genuinely comprehensive knowledge, however." Yes. We should abandon expectations of comprehensive knowledge.

79 "The system is designed to be capable of at least expressing all properties of its objects--" We should be careful about using the term 'all'.

91 "I think ...that what is at stake in all of these cases is in fact the *same* difficulty, though appearing in slightly different guises—the suspicion that there is a single deep problem regarding truth, knowledge, and totality that is simply visible in different forms in the Liar, the Knower, and Gödelian results." Yes. And IMHO that single deep problem lies in the acceptance of the notion of infinity.

91 "The proof that there can be no set of all truths is as follows. Suppose there is a set of all truths *T*: *T* = {*t*_{1},*t*_{2},*t*_{3},...}" You are assuming a countably infinite set of truths.

97 "There can, for no world, be a set of *all* truths that hold in that world." This result seems to render as ambiguous the very notion of "all" and the notion symbolized as "Ɐ".

98 "This much, at least, seems to be a solid result of the work above: there can be no set of all truths." Yes. So we need a more precise definition of 'all'. E.g. all that is known by x, all that can/will be known by x, all ____ extant at some time *t*, etc. And to me it seems to imply a finite dynamic world.

98 "Within naive set theory one all too quickly encounters the contradictions of Cantor's paradox regarding a set of all sets and Russell's paradox regarding a set of all sets not members of themselves."

98 "*All* axiomatic set theory, standard or alternative, is essentially a response to these two paradoxes."

100 [T]he central strategy of each of these [alternative] systems is to escape set-theoretic paradoxes by supplementing sets with an additional category of nonset classes."

101 "Specker (1953) went on to show that the non-Cantorian sets of NF [Quine's New Foundations] cause the relations of lesser to greater among cardinals to fail to be a well-ordering, and he thereby a [sic] produced a disproof of the axiom of choice within NF."

101 Grim quoting Quine: "The fact remains that mathematical induction of unstratified conditions is not generally provided for in NF. This omission seems needless and arbitrary. It hints that the standards of class existence in NF approximate insufficiently, after all, to the considerations that are really central to the paradoxes and their avoidance." I don't think the omission is either needless or arbitrary. Induction brings with it an unwarranted assumption that does not hold in finite systems. We should therefor reject the notion of infinity.

101 "Quine proposed the system of *Mathematical Logic*. ML is basically an enlargement of NF so as to include a category of *ultimate classes* in addition to sets." This sounds like an excellent approach to me.

101 "With both classes and sets on hand in ML it is important to distinguish between *V* a "genuine universe" that includes all classes, and U*V*, which includes merely all sets." This seems to make even more sense if *V* is finite.

149 "Hao Wang charges, on the basis of a result attributed to Rosser, that if NF is consistent, the class Nn of natural numbers cannot be a set within ML (Wang 1986, 640). As Quine notes, however, the real Rosser result is significantly weaker: that if NF is consistent, within ML Nn cannot be *proven* to be a set (Quine 1986)." I think Wang is right. The natural numbers are capped.

105 "[A]s in the case of NF, the costs of seriously adopting something like VNB turn out to be exorbitantly high. Moreover, here, as in the case of NF, a major cost is again a sacrifice of mathematical induction. As Lévy notes, drawing on work by Mostowski,'' A particularly embarrassing fact about VNB is that in VNB ... one cannot prove all instances of the induction schema, ..." I say we should not be embarrassed. Instead we should drop induction or admit it only up to Practical Numbers and develop mathematics on that basis. I think it will be practically the same for analysis and for all finite systems.

106 "Despite these important features, however, it turns out that KM-like [Kelley-Morse] theories don't ultimately offer any greater hope for a set or class of all truths than do the alternative theories considered above." The problem with this concern is that it is based on an imprecise definition of the word 'all'. With a proper definition, the concern will evaporate.

106 "Here as before, the natural course seems to be to expand KM to a KM-like system incorporating classes of truths by introducing a correspondingly unrestricted comprehension principle for classes of truths. ...The lesson of alternative set theories considered so far—NF, ML, VNB, and KM—seems to be uncompromisingly negative regarding prospects for any collection of all truths." That may seem to be "the natural course", but to me it is like trying to get out of a hole by digging it deeper. Much better to get out of the hole and abandon that approach. Let's give up on the notion of infinity.

107 "[T]he cost of such restricted comprehension seems to be a corresponding and strongly counterintuitive crippling of mathematical induction as well. Such limitation, as Quine notes, seems "needless and arbitrary"" I disagree with Quine. The loss is not counter-intuitive to me but just the opposite. The notion of induction is needless and its acceptance is arbitrary.

107 "Ackerman set theory", ..."

107 "A is thus a "class-down" theory, in a sense, rather than "set-up"; within A we have sets as classes for which the primitive *M(C)* holds, rather than classes introduced as additional entities "on top of " sets." Seems to be a reasonable approach to me.

109 "None of the alternative set theories surveyed, then, seems to offer any very promising route of escape. The technical problem that has dogged our steps throughout is basically this. An *un*restricted principle of comprehension for classes—such as that in KM, ML, or A—if extended to give us classes of truths, allows too much of the basic mechanism of a Cantorian argument to permit any hope for a class of *all* truths. Comprehension restricted to stratified or predicative conditions, on the other hand, such as in NF and VNB, *does* seem to promise the possibility of a class or set of all truths precisely because it cripples the basic mechanism of the Cantorian argument. But in each case the Cantorian argument is crippled only at the cost of seriously crippling mathematical induction as well." That's OK by me.

150 "In general, of course, it can also be argued that alternative set theories demand a cluttered and less natural choice of primitive notions. Indeed, it's far from clear what intuitively distinguishes classes from sets, or why we should think there is a second level of collections above and beyond collections offered by sets." Well, how about this: Let subjectivity/objectivity distinguish between classes/sets. That involves humans and human judgment but after all, that is our world. The "second level of collections" could be interpreted epistemologically as separating facts from theories, or theories from speculation, or even known from unknown. Or Practical Numbers from defined numbers.

110 "With an eye to the issue of omniscience, consider also the class of classes of things known by existent beings. Wouldn't what God knows be a member of *that* class?" No. Not if we deny God's omniscience, which I think we should, and which seems to be Grim's inescapable conclusion.

110 "Quine's general objection: "[VNB modified] shares a serious drawback with ML, and with von Neumann's unextended system, and with any other system that invokes ultimate classes. ... We want to be able to form finite classes, in all ways, of all things there are assumed to be ..., and the trouble is that ultimate classes will not belong"" We simply need to refine the definition of 'all' in an intuitive way.

111 "But standard ZF set theory minus the power set axiom, he [Menzel] suggests, *does* offer such a possibility," I wonder what that "power set axiom" is. I never heard of it. I wonder how it relates to the Axiom of Choice (AC). It seems to me that this approach might relate to my Practical Number idea.

111 "[T]here is no way of *generating* basic infinite sets within ZF – Power at all;" IMHO that should be acceptable. After all we never use infinite sets or even real numbers for that matter in any human activity.

113 "The problems at issue don't appear to be merely surface features of some unfortunate symbolism. They seem rather to be difficulties embedded in our notions of truths, propositions, possibilities, and totalities themselves—difficulties that any system that adequately captures such notions will have to reflect, and that no simple appeal to alternative systems is going to dispel." Yes!

115 "How serious such a difficulty really is depends, I think, on what we expect of formal semantics. If formal semantics is thought of as an attempt to explicate the meaning of our symbolism or to indicate how that symbolism is to be understood, this does indeed seem a major difficulty. ...But if formal semantics is viewed as something significantly weaker, perhaps as the mere construction of formal models of *some* features of our symbolism, adequate for *some* purposes, this difficulty may not be so telling." If we consider this question in the context of Rosenberg's NI hierarchy, or in Beon Theory, then the second case seems to obtain. In both cases, reality is finite.

116 "Formal modeling in general may offer a poor shadow of either the meaning of our symbolism or our understanding of it, and thus the failure of such formal modeling for quantification need not reflect any failure in our general understanding of its meaning." Yes, consider the meaning of numbers thousands of years prior to Peano.

116 "[P]utting sets aside, can't we speak quantificationally of a *property* shared by all and only those propositions that are in fact true?" I agree with Grim's conclusion that the answer is "No", but I think the trouble starts earlier. I think the trouble is that the word 'all' cannot be given a consistent definition and is therefore meaningless. This carries into formalism by making the symbol 'Ɐ' meaningless. Consider this construction: All stars. Suppose we were to make some assertion or hypothesis involving "all stars that have not exploded". Which ones, exactly, would that include? How about those whose pre-explosion starlight is just now reaching us? If we attempt to establish a cut-off "now", how could we in the face of the non-existence of simultaneity? Closer to home, what would be included by the phrase "all dogs"? Consider animals of mixed breed, or slowly dying, or in various stages of birth. I think the very concept of sets has fuzzy edges. Or consider S.J. Gould's question, "What, if Anything, is a Zebra?

118 "It might be charged, however, that we are still implicitly smuggling set-theoretic notions into such an argument by way of the crucial concepts of mappings or functions, one-to-one correspondences, and cardinality." Yes. I, for one, would make that charge. While Cantor's definition of one-one correspondence works fine for finite sets, it fails for any putative "infinite" set. Consider Feynman's observation that "There are more numbers than numbers."
Or consider my argument involving convergent vs. divergent error during the "correspondence process." In my argument, the "error" is the answer to the question, How many elements are missing from your correspondence so far? As the correspondence process proceeds, the error will go to zero in the finite case but increase without limit in the "infinite" case. E.g. How many numbers are missed so far in making a 1-1 correspondence between natural numbers and even natural numbers? Well, the answer is half the number of numbers less than the last even one put into the 1-1 correspondence. That error increases without bound, although it is always finite.

118 "To avoid the Cantorian arguments of preceding sections, it appears, we need to renounce not only explicit reference to comprehensive collections of all truths or all propositions but also any reading of a formal semantics for quantification that would implicitly resurrect them." Yes. I still say that it is the use of the word 'all' that is the problem.

121 "What this paradox offers, I think, is a glimpse at the conceptual core of the Cantorian argument we've pursued throughout. All concepts crucial to both sides of the argument appear already in the question it addresses: can all propositions be put into one-to-one correspondence with themselves or not? Once given the resources to ask such a question, it appears, we have all we need to force us to paradox." Yes. I agree. But in my opinion the crucial "resource" that is needed is a clear notion of the word 'all'. If the notion includes an "infinite" number, then paradox is unavoidable. But if 'all' refers to those in 1-1 correspondence with the Practical Numbers (my definition) then paradox can be avoided. IMHO.

122 "What the arguments above ultimately indicate is *not* merely that all truths, somehow unproblematically referred to, fail to form a set. What they show, on the contrary, is that the very notion of *all* truths—or of all propositions or of all that an omniscient being would have to know—is itself incoherent." Yes! Exactly! The very word 'all' and the symbol 'Ɐ' are incoherent, or at best ambiguous. The term is both context dependent and knower dependent.

123 "There is no "set of all truths," nor for that matter any coherent notion of "all truths," much as there is no such thing as "the square circle" or "the largest positive integer."" I agree. But I claim that the notion of 'all' can be made coherent by specifying a knower and stipulating a context. All facts known by a certain knower can make sense just as a square circle can exist in Abbott's *Flatland* context as a cylinder of radius r and height 2r. When introduced to A. Square with its axis in A. Square's plane, it appears as a square, but when rotated 90° about a line in A. Square's plane that is perpendicular to the axis, which of course requires an additional dimension for A. Square, then the same object appears as both a circle and a square to A. Square. This is similar to a largest positive integer making sense in the context of my Practical Number system.

123 "In each case what we deny is that there is anything that fits a certain description, a description that we would contradict ourselves by straightforwardly *using*." We have to be careful to recognize that in each case the denial is only effective in a certain context. That is usually tacitly assumed but would be better explicitly stated.

123 "The logical core is simply that the supposition of an omniscient being, the postulation of a set of all truths, or the assumption of a proposition about all propositions leads directly to contradiction." That is assuming that the claim is being made for "all" contexts. But what are "all contexts"? Grim has demonstrated that there is no such thing. So all claims can only be made in specified, limited, contexts.

123 "[A]t the point we are tempted to speak of "all truths" or "all propositions," we already face incoherence." I agree. But again, it is because a context-free notion of "all" is incoherent.

124 "[I]t appears [that] the universe refuses to be *either* a one or a many." Yes! We need to re-think the very notion of "universe"

127 "My arguments throughout have clearly employed, and in that sense clearly rely on, contemporary logic as we know it." You have also relied on the Gödelian argument, which in turn relies on an infinitude of integers. Cantor showed that this assumption, that of the existence of an infinite set, leads to contradiction. Over a hundred years of attempting to sweep the resulting inconsistencies under the rug has failed, as you so expertly show. It is time to abandon that assumption and acknowledge that reality is a finite, extended, evolving work in progress.

127 "Perhaps, despite appearances, it would be possible, for example, technically to specify a "gob" or a "bunch" that would both coherently and intuitively collect or cover "all truths" in a way that neither sets nor classes nor even raw quantification evidently can. Perhaps." I suggest that instead of redefining 'set' and 'class' we redefine 'all' to mean all in a particular specified context.

128 "Perhaps our logic is merely inadequate to do justice to real circular squares or married bachelors." It is not an inadequate logic that is the problem. It is the failure to specify the context. For example, in the context of *Flatland* a real circular square can exist as a cylinder of radius r and a height of 2r as seen by A. Square viewing different intersections. Or, in the context of a culture that allows gay marriage and in which "wife" is defined as a married female and in which 'bachelor' is defined as a man without a wife, then indeed a married bachelor makes sense.

129 "Standard and nonstandard arithmetics ultimately offer divergent views of what numbers *are*, and our conception of what *arithmetic* is has inevitably changed as a consequence. The mere existence of alternative geometries has made the enterprise of geometry itself a different discipline—a matter no longer of intuiting a priori essential properties of space but of tracking the implications of speculative spatial hypotheses." Yes, exactly. That is why I think Practical Number theory can significantly enhance our "speculative [spatio-temporal] hypotheses".

129 "[These speculative hypotheses] might come to be valued as something more: creative modes of conceptual freedom." I agree.

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