One should not forget that even logical laws are doubtable whenever they are interpreted extensionally as referring to a collection of real-world objects, as opposed to when they are used intensionally as rules of production.

Most of the time logic is used and appealed to directly, without reference to extensions, because it is used for asserting normative statements. The logical description of an ideal electronic circuit is comparable to the expression "Tidy your room! because i said so!". This ideal use of logic is comparable to ethics and it makes no sense to speak of epistemic doubt here. In contrast, if a circuit description was thought to universally describe the operations of real world electronics, this is obviously a highly doubtable proposition.

Physical laws are the joint expression of normative sentiment and physical description and so they aren't pure propositions in the ideal philosophical sense. The normative part is expressed by the use of universal quantifiers that aren't falsifiable and which say "Every X is a Y". But they don't need to be falsifiable, for their purpose is political,namely to assert scientific and cultural policy in the same way as the electronic circuit design that implicitly asserts "Intel should make chips this way".

Getting back to the original discussion, consider how one determines measurement precision. Isn't it's very definition ultimately in terms of the reproducibility of experimental results? In which case, if repeated experiments fail to reproduce results, then by definition measurement precision is lacking.

So it’s an output tool, like a painting on canvas or printed pages. But what is the vision in feeding AI words and lines from existing poems? There is no vision except to create what is now redefined as a poem, as art. There is no poet, only the programmer. The vision then becomes that of the reader, as in reader-response literary theory. There is no vision of the artist because the construction of the poem is random and the meaning accidental. — Brett

Modern creative algorithms are accessible to anyone, and have many tuneable parameters that the artist himself can control in accordance with his artistic vision, to influence the style, subject etc . But perhaps this is tangential to the discussion.

Is the creative process a self-directed and inwardly driven process determined by the artistic foresight of the visionary artist? or is it a blind and partly external environment-driven process, whereupon external stimuli bring about artful stimulus-responses in a person said to have artistic temperament?

I suspect that Cartesian minded internalists who believe in the former might have a harder time reconciling AI and the arts compared to externalists who view the artist as a shambolic director of
haphazard external processes.

sime

Art algorithms ... accelerate the pace of art revolutions.
— sime

Can you give evidence for that? — Brett

Art has moved in tandem with the accelerating technological trajectory that began with the invention of electronics in the Victorian era. Without semi-autonomous content creation tools, the present creation of vast and open virtual worlds wouldn't be possible. Perhaps we could call the potential algorithmic output of content-creation tools "meta art" in comprising a distribution over art objects, but it is still 'art' in the traditional sense in being ultimately shaped by the vision of it's users who program it, feed data to it, and tweak it's responses. And these tools also complement and fuel the need for traditional artists who produce hand-crafted content, as for example in virtual world content creation where there are neither enough algorithms nor artists to produce the infinite amount of diverse content required.

Artistic inspiration invariably refers to an external source of information and stimulation for the artist from where they get their ideas. A poetry generating algorithm is just another source of artistic inspiration, whose output a poet will tweak to his own satisfaction. This is little different to the use of 'story dice' as random source of inspiration.

In my opinion, AI art does not represent a paradigm shift in terms of the meaning of art, rather it reveals a significant part of the preexistent process called artistic inspiration and democratises it. Art algorithms are really about increasing the economy of scale of art production and they accelerate the pace of art revolutions.

If Presentism does not entail the reality of passage (i.e. the A-Theory), then are you arguing for the position of Presentism + B-Theory, i.e. that only present objects exist and that temporal passage is not real? I have never heard of this before. This is like the converse of the Moving Spotlight theory (Eternalism + A-Theory). I can only note this is at odds with the definition of Presentism given in most places, including the SEP article on Presentism:

presentism can be understood as the following conjunction:

(PC) (i) Only present things exist,
& — Luke

Yes, in my view the logic of presentism, or at least what I call presentism, leads to a reduction of the so-called "A series" into a perspectival interpretation of the B series, that is fully coherent with the best scientific theories, including special relativity for all empirical and practical purposes. The reason why I call a perspectival interpretation of the B series "presentism", is due to the fact that tenses are treated as indexicals, where an indexical can be considered to be an act of pointing to something, where the "something" is empirically undefined up until something actual and specific is pointed at.

In fact, i'd consider presentism to be the temporal logic of perspectivalism. From wiki

Perspectivism (also perspectivalism; German: Perspektivismus) is the view that perception, experience, and reason change according to the viewer's relative perspective and interpretation. It rejects both the idea of "one unchanging and essential world accessible to neutral representation by a disembodied subject."

I also consider the entire work of Wittgenstein to be a commentary of the logic of presentism, starting from the Tractatus that reduced every proposition including tensed propositions to observable empirical relationships among present atomic elements. See Hintikka for more on Wittgenstein's implicit philosophy of Time. The philosophical investigations is also useful for explaining the conflicting intutitions between presentists and growing block enthusiasts; presentists point out that "past" and "future" are actually used as empirically undefined indexicals, as opposed to growing block enthusiasts who think of the meaning of "past" and "future" in terms of mental imagery they assign to those notions, mental imagery which they overlook is part of the very present.

I notice that Wiki has also mentioned a similar criticism of the "spotlight/growing block" theory from the perspective of indexical tenses. In fact its the only criticism wiki documents against the growing block theory:

https://en.wikipedia.org/wiki/Growing_block_universe


Can a growing block enthusiast explain to me what explanatory value their theory of time adds to a present observation of a physically growing block? What is the nature of the "meta-time" that this growing block of time must have in order to grow time? and what about the meta-meta-time needed for that and so on?

But yesterday was a different day to today, just as tomorrow will be.
. — Luke

According to the indexical theory, to say "tomorrow will become yesterday" is to merely to express an intention to redefine the meaning of "yesterday", "today" and "tomorrow" so as to give the illusion of temporal passage. In other words, the indexical theory is an anti-realist stance. It's relevant to the issues you raise, because the use of tenses as indexicals is routinely overlooked in philosophical discussions.

You say that a denial of passage need not involve a denial of the past and future, but if "the state of the river is also our notion of "the present", then isn't this a denial of past and future? This seems to imply that we have no 'notion' of past or future states by which to judge that the present has changed. — Luke

All that is being denied is a notion of "temporal passage" that is distinct from the passage, of say, of a speeding train relative to the readings on a stop-watch. In other words, that the notion of temporal passage reduces to relations among appearances, which can also include whatever experiential content one has temporarily assigned to the notions of "past" and "future".

If understood indexically, the past is always the past and the future is always the future, for yesterday is always yesterday, and tomorrow is always tomorrow....

I tend to agree that a true presentist who rejects the existence of the past and future would be unable to judge which time is present. However, in reality, I think we are all able to ascertain this and can talk meaningfully about temporal passage. But this is not the focus of this discussion — Luke

For similar reasons I disagree that a denial of passage of time involves the denial of past and future, since "past" and "future" can similarly be interpreted as indexicals.

We can say that the state of the river has changed relative to the state of a photograph. But if the state of the river is our notion of "the present", then we can no longer say that the river has changed relative to the present.

I believe that McTaggart was making a similar deflationary argument when he concluded the unreality of the A series.

I disagree that presentism entails the reality of passage, because presentism might interpret the word "now" as being an indexical that cannot refer to the same set of affairs twice. If that is the case, then temporal passage cannot be referred to.

Recall Heraclitus, who said that it is impossible to step twice into the 'same' river twice. Here he is implying that the meaning of "same river" refers to a constant, say a static memory, relative to which the state of the actual river can be said to have changed. In contrast, if it is denied that the meaning of "the present" is fixed, then the river cannot be said to have changed relative to the present.

If the phrase "the present" is always substituted for the current international atomic time, then the sentence "the present has changed" is no longer grammatically permissible.

To many enthusiasts, panpsychism isn't so much an explanatory theory of consciousness, but an Occam's Razor style argument that non-living systems should be considered to have identical metaphysical properties as living systems, on the basis that there is no falsifiable justification for considering their metaphysical properties to be different.

From this perspective, pan-psychism is in a logical sense very close to if not indistinguishable from eliminative-materialism, the difference being that panpsychism doesn't consider subjects who claim to possess consciousness as being factually false, but as being necessarily and vacuously true in virtue of consciousness being a universal and hence tautological property. From this perpsective, the main difference between panspsychism and eliminative materialism is optimism.

When it comes to general epistemological questions of the form 'Can science study X'? the answer depends on the extent to which X is considered to constitute the very meaning of scientific practice. In the event that X is considered to ground the meaning or truth conditions of scientific practice, science can only be said to study X if science is considered to be it's own meta-science. But that assumption in turn raises worries and doubts as to the consistency, meaningfulness and reliability of the consequently circular scientific epistemology.

Consider similar questions: Can and to what extent can science study causality? or the existence of space, time and phenomena? or the reliability of epistemology suitably naturalised? can it even be meaningfully asserted that science can study the cosmos?

I don't think someone could be very functional having their self identity undermined as we see in cases of amnesia and dementia. It is useful to keep track of who you are and exhibit a consistent personality. — Andrew4Handel

In other words, the "self" is a useful idea with practical utility. But does that warrant the promotion of the "self" to the status of ontological primacy?

Most everything you believe has come from the testimony of others, if you doubted most of it you would be reduced to silence. Professors, books, language, science was given to you by others, you probably had little to do with creating the information yourself. — Sam26

I'm specifically referring to the trustworthiness and reliability of the verbal reports of experimental subjects in psychological experiments where they are tasked with giving self-reports, possibly including explanations for their own behaviour. A testimony of a subject taken at face value can be terribly misleading when it comes to understanding the actual underlying proximal and distal causes of the subject's verbal behaviour, for there is no reliable mapping between a person's use of sentences and their psychological state, and people don't possess introspective access to the causes of their own behaviour.

I don't even trust personal testimonies when it comes to deciding the veracity of the humdrum theories of behavioural psychology, let alone for deciding the veracity of pseudo-scientific mystical hypotheses.

That said, i have sympathy with the sentiments expressed by beliefs in "life after death"; not in the sense of it constituting an empirically contingent and testable scientific hypothesis, but because the opposite notion of 'eternal oblivion' is equally nonsensical.

These are very interesting remarks. Sadly, my knowledge of dynamic logics is sorely lacking at this point in time, but I think dynamic logics at best can only have partial applications; for there are many cases where we need to use a static logic. And it is in these scenarios that the Liar Sentence arises. — Alvin Capello

Certainly the semantic contradiction arises when the meaning of the liar sentence is analysed statically, but there is nothing that necessitates this adoption of tenseless logic in either the construction or analysis of liar sentences.

Indeed the construction of all proofs is a dynamic process over time. In the case of the liar sentence, a typical verbal explanation of the paradox involves alternatively saying "I am telling the truth about my lying, therefore I am lying about my lying, therefore i am telling the truth about my lying...etc". What is static in the construction of this paradox? Isn't the insistence that the liar sentence must be understood statically, the source of the contradiction?

Surely some seeming contradictions can be resolved, but I don’t think this is true of all of them. For instance, I don’t think the Liar Sentence and other similar semantic paradoxes have any consistent solutions, so these are radically contradictory objects on my view. — Alvin Capello

Of course, liar paradoxes are only contradictions if their truth is considered to be atemporal; otherwise these contradiction are avoidable using a tensed logic in which every sentence of a proof is temporally indexed according to the moment of it's creation, wherein the only distinction between premises and conclusions is that the latter is constructed after the former.

In such a tensed logic, liar paradoxes of the form P(t) => ~P(t+1) are consistent and only the simultaneous derivation P(t) and ~P(t) is inconsistent.

There's a difference between using words to denote objects or relationships between objects in the world, and the objects and relationships between objects in themselves that those words represent. The Law of Non Contradiction is thought to be violated only because it can be shown that a contradiction in terms of the relationships between the symbols (i.e. words) that point the objects, can be true. This results for the false equivalence that the symbols that represent objects and the objects themselves are the same, or rather, have the same logical form which they do not. In an actualized sense, nothing can ever exist and not exist at the same and in the same respect. However, in a state of potentiality, the actualized possibility of x and -x exist at the same time and in the same respect, according to my philosophy anyways. — TheGreatArcanum

I agree that contradictions are properties of sentences rather than of matters-of-fact, for I cannot understand what could be meant by contradictory matters of fact. I would also say the same about truth, for I cannot fathom a false matter-of-fact. The principle of non-contradiction is certainly critical to the practice of science, but I see neither justification nor practice of non-contradiction when it comes to philosophy.

Its not about logical atomistic consistency. Wittgenstein should not be in your list sime. — Gregory

I wasn't specifically thinking of logical atomism, i was referring to his consciously self-refuting Tractatus, as well the latter Wittgenstein's philosophical investigations, that isn't logically consistent. For example, his apparent reliance on the imagination to refute the idea of private language. This isn't a criticism, it's just a general feature of philosophical arguments. For many other examples see Graham Priest's "Beyond the Limits of Thought".

PNC is either rejected or violated in the works of many philosophers, e.g. Heraclitus, Kant, Hegel, Wittgenstein... There isn't much evidence to support the logical consistency of philosophy, especially in epistemology. If philosophy is considered to be primarily a normative activity, this doesn't matter. The loss of PNC isn't a great blow, it just means that philosophers are unstable hypocrites with alternating beliefs.

If we are constantly changing our opinions as to the facts of the past on the basis of new information, then why should we believe that the past is real and immutable?

Suppose that in 2030 society obtains decisive historical evidence concerning the identity of Jack The Ripper in 1888, whereby historians thereafter claim that the riddle regarding Jack the Ripper's identity was solved in 2030. Why should we believe that the actual facts regarding the identity of Jack the Ripper in 1888 existed before 2030? What does this assertion add to our calendar-indexed observations?

The deeper problem concerns the fact that in Cartesian co-ordinates the notion of vertical and horizontal Euclidean lengths is incommensurate with the notion of diagonal Euclidean lengths; hence the reason why a hypotenuse that is diagonal to the Cartesian axis is assigned an irrational number such as Sqrt(2), which of course isn't a quantity but a non-terminating algorithm for generating a Cauchy sequence.

As 180proof mentioned, If atheism is identified with the absence of belief, then it avoids the 'truth-by-correspondence' problem concerning beliefs that have non-existent referents. However, this is arguably not the case for theism, that your argument can be turned around to defend, by the following argument:

Premise 1 : All beliefs have referents.
Premise 2: Theism is a belief.

Conclusion: The referent of Theism exists, and therefore theism is true.

Personally, I find this argument acceptable, because the idea that a non-existent object can cause belief-behaviour is scientifically unacceptable, leading me to the conclusion that all beliefs are vacuously true in the epistemological sense of truth-by-correspondence of language to something. Hence any substantial notion of truth cannot be in terms of "truth by correspondence" of language to reality, but in terms of ethics and cultural convention.

A mistake of atheists is to assume that the object of theistic beliefs is universal, for there are many potential physical causes of religious behaviour and speech.

Well obviously the reason why people enjoy playing Chess is because its outcomes are uncertain due to players bounded rationality and tendencies to make mistakes, assuming that the skill difference between opponents is roughly even. This is especially the case for the variant Chess960 that in being randomly initialised diminishes the role of opening-theory. The Chess community are well aware that the rules of Chess have to evolve if Chess is to remain an interesting non-predetermined spectacle. Perhaps the game will continue to fragment into more and more alternatives. Personally, I think there are more interesting board-games to professionalise.

Irrespective of dualism, it isn't clear in any case what is meant by physical interaction, due to conflicting opinions as to the metaphysics and existence of causality. If one goes so far as to deny the literal existence of counterfactuals then interaction isn't even a substantive concept. Therefore ontological dualism and more generally, ontological pluralism, don't necessarily imply interaction problems, but only that different descriptions of the world cannot be inter-translated.

All working physicists informally appeal to "directness" whenever they make an inference, even though Physics possess no theory of directness. For otherwise a physicist could not claim to learn anything from an experiment, nor for that matter could he find the sentences of physics intelligible. So although directness/indirectness aren't themselves defined in terms of physical criteria, the converse is true.

Exactly the same issue applies to language in general, for we are taught the meaning of words either through ostensive definition, or by verbal definitions that implicitly appeal to earlier ostensive definitions for their intelligibility. And yet we have no linguistic criteria for translating verbal definitions into ostensive definitions and vice-versa, for languages are only publicly defined up until verbal criteria.
.
In line with language in general, the semantics of Physics is both under-determined and redundant; one Physicist's "natural" object is another Physicist's "metaphysical garbage", because they might each understand physics using different semantic foundations that are rooted in different ostensive definitions.

Everything is 'true' is a position I independently arrived at, without knowing that this epistemological position already existed under the banner of Trivialism.

Essentially, trivialism (at least as I am using the term) says that every belief is seen to be true once the object of the belief is identified with its immediate causes. Trivialism is a corollary of semantic deflationism and presentism, which denies that a prediction can actually refer to a future event by virtue of the future not existing in a literal sense in being a mere indexical.

For example, suppose that Alice becomes convinced that she will win the lottery and buys a ticket. According to the causal theory of reference, her belief that she will win the lottery is nothing other than a report referring to her immediate situation. If in fact she doesn't win the lottery, then according to trivialism she is only said to be "wrong" by reinterpreting the object of her belief to refer to the results of the lottery via a post-hoc revision of linguistic convention.

The collective use of language constitutes an inconsistent convention, for everybody uses the first-person pronoun to refer to a different subject. This is the central oversight in debates over idealism and realism that entirely ignore who is making an ontological commitment, such as the existence of colour.

Ordinarily, if I assert "I am seeing a red apple" the meaning of the sentence cannot be decomposed into two independent assertions, namely one of a subject and another of an object, as is in situation where I assert that someone else seeing a red apple. As far as I'm concerned, red, i.e. my red, exists independently of other people's perceptions of my red, and they cannot possibly know this fact, for whenever they talk about red they are referring to their red. And the situation isn't improved by talking only about "objective" optical properties.

Therefore consider the irrealist alternative; namely that ontological disagreements are partly the result of our collectively inconsistent use of language.

'Phenomenal red' is an estimator of 'optical red' in common situations. No necessary relationship between phenomenal colour and optical qualities can be defined nor established, due to the impossibility of exhaustively specifying and testing their relationship.

First start with the notion of material-implication:

Classically, A=>B means that if A is true then B is true,and is equivalent to NOT A is True OR B is True.
Constructively, A=>B only means that a proof of B can be derived from a proof of A, and says nothing about the actual truth or provability of A or B.

The classical interpretation of material implication would say that you cannot be a lizard because you don't like flies, which shouldn't be problematic to assert, assuming that we live in a closed world containing a finite number of lizards that we can count in order to check their taste for flies.

But in the event we live in an open world containing a potentially infinite number of lizards, the classical interpretation runs into a problem in that the truth of A=>B can never be verified, implying that A=>B can never be asserted. And yet we do use conditionals without assuming that we live in finite closed worlds, which indicates our actual use of material implication is constructive rather than classical. For example, our definition as to what a lizard is includes the fact it eats flies, and therefore A=>B becomes somewhat tautologous.

Because I have not seen any resolution to these questions, I would not say that a "rule" has any existence at all. — Metaphysician Undercover

I would say that the laws of Mathematics and Logic are normative principles pertaining to conduct regulation so as to make the world easier to describe and manipulate.

These normative principles cannot be given a logical justification on pain of circularity, rather their justification stands or falls with their general overall usefulness.

In your view, what is a belief, and how should the object of a person's belief be ascertained?

Should the object of a person's belief be identified with the physical causes of their belief, in which case every belief is seen to be necessarily true when it is physically understood, or should the object of a belief be decided impersonally by linguistic convention so as to reflect the normative values of the person's community?

Wait so you just made that up? It's not a real thing? You had me convinced. Why not mod out the reals by the trivial ultrafilter and see what you get? What do you get?

Why are there so many die-hard constructivists on this forum? If you go to any serious math forum, the subject never comes up, unless one is specifically discussing constructive math. You never see constructivists claiming that their alternative definitions are right and standard math is wrong. Only here. It's a puzzler. — fishfry

Well obviously from a pure mathematics perspective, every proof in ZFC is considered construction, in contrast to Computer Science that has traditionally had more natural affinity with ZF for obvious reasons, and there is a long historical precedent for using classical logic and mathematics. As a language, there is nothing of course that classical logic cannot express in virtue of being a "superset" of intuitionistic logic, but classical mathematics founded upon classical set theory IS a problem, because it is less useful, is intuitively confusing, false or contradictory, lacks clarity and encourages software bugs.

In my opinion, Constructive mathematics founded upon intuitionistic logic is going to become mainstream, thanks to it's relatively recent exposition by Errett Bishop and the Russian school of recursive mathematics. Constructive mathematics is practically more useful and less confusing for students in the long term. Consider the fact that the standard 'fiction' of classical real analysis doesn't prepare an engineering student for working in industry where he must work with numerical computing and deal with numerical underflow.

The original programme of Intuitionism on the other hand (which considers choice-sequences created by the free-willed subject to be the foundation of logic, rather than vice versa) doesn't seem to have developed at the same rate as the constructive programme it inspired. However, it's philosophically interesting imo, and might eventually find an applied niche somewhere, perhaps in communication theory or game theory.

BTW, i'm not actually a constructivist in the philosophical sense, since the constructive notion of a logical quantifier is too restrictive. In a real computer program, the witness to a logical quantifier isn't always an internally constructed object, but an external event the program receives on a port that it is listening. What's really needed is a logic with game semantics. Linear logic, which subsumes intuitionistic and classical logic is the clearest system i know of for expressing their distinction and their relation to games.

As for a trivial ultrafilter, its an interesting question. Perhaps a natural equivalence class of Turing Machine 'numbers' is in terms of their relative halting times. Although we already know that whatever reals we construct, they will be countable from "outside" the model, and will appear uncountable from "inside" the model.

Now that's something I've never run across. Both too big and too small at the same time. But it takes a weak form of the axiom of choice to have a nonprincipal ultrafilter, which is needed to construct the hyperreals. Do constructivists allow that? — fishfry

emmm......... Nope :) for the reason you've just mentioned. For where is the algorithm of construction? Of course , the trivial principle ultrafilter is permitted, which then produces a countable model..

By "constructive hyperreal" i was merely colloquially referring to using functions such as f(n)=1/n as numbers according to some constructive term-oriented method that didn't involve assuming or using cauchy limits.

Now obviously, any countable list of Provably Cauchy-Convergent Total Functions is unfinished, in the sense that a further PCCTF can be built that is is not already in the list via a diagonal argument. No problem, we just shuffle along the existing enumeration to add the new function into the existing list. But then doesn't this contradict the notion that our previous list was complete?

There seems to be an ambiguity between two definitions of completeness. If Dedekind completeness is understood to be an axiom of construction then it is trivially satisfiable in the sense that the axiom itself can be used to assist in the generation of a real from an existing list of real numbers. After all, if there wasn't a countable model of the Axioms of the reals, then they would be inconsistent, since Second-order quantification can always be interpreted as referring only to the sets constructively definable in first-order logic.

On the other hand, if completeness is understood to refer to a finished list of PCCTFs, our list is not complete in that sense.

So it seems to me that countable model of reals, both first and second order, are especially useful ( not to mention the only models we use in practice),for clarifying the relationship between Dedekind completion, Cantor's theorem and ordered fields.

If one abandon's the second-order completeness axiom, and possibly cauchy convergence, then there are less constraints in the construction process, allowing one to define a potentially larger field of computable numbers that includes infinitesimals as is done with the (constructive) Hyperreals, and one can even include computable 'numbers' that are aren't provably total. In which case ones countable list is now finished, but now there are no more numbers to be added, because now the diagonal argument cannot be used to construct a new numbers in virtue of one's list including non-numbers that aren't guaranteed to halt on their inputs.

So i hope this had lead to a satisfactory conclusion.

The computable numbers are countable. That's because the set of Turing machines is countable. Over a countable alphabet there are countably many TMs of length 1, countably many of length 2, etc.; and the union of countable sets is countable. QE Freaking D. — fishfry

?? Perhaps I should have been clearer from the beginning, but i took everyone's understanding for granted that a computable number refers (in some way) to a computable total function. Apologies if that is the case. For surely you appreciate that the computable total functions aren't countable?

The computable total functions are a proper subset of the computable functions that also contain partial functions. i.e. that do not halt on a given input.

It is true to say that the whole set of computable functions is countable, for reasons you'e sketched. It is not true to say that the set of computable total functions are countable, for we cannot solve the halting problem. Hence the reason why we say the computable numbers are sub-countable: the only way we could 'effectively' enumerate the computable numbers is to simulate every Turing machine and wait forever, meaning that any 'candidate enumeration' we construct of our computable numbers after waiting a finite time is also going to contain computable functions that aren't total and hence are not numbers.

For the constructivist, this "subcountability" is all 'that 'uncountability' means. It is simply means that we can never construct a total surjective function from the natural numbers onto the computable numbers. It doesn't mean in any literal sense that we have more computable real numbers than natural numbers.

The sequence of n-th truncations of the binary expansion of Chaitin's number is a Cauchy sequence that does not converge to a computable real. End of story. Then you say, "Oh but that sequence isn't computable," and I say, "So freaking what?" and this goes on till I get tired of talking to yet another disingenuous faux-constructivist. — fishfry

We have to be careful there. We can run every Turing Machine and at any given time create a bar-chart of the ones which have halted, and this histogram comprises a sequence of computable functions whose limit isn't a computable function. To my understanding this sequence of functions isn't cauchy convergent, for we cannot construct a bound on the distance between successive histograms. Let's not forget that there are an infinite number of computer programs of every size.

Compare this situation to a computable total function f(n) representing the "values" of the Goldbach's Conjecture; Let's say that f(n) = 0 if every even number less than n is the sum of two primes, otherwise f(n)=1. Here we can also compute the individual digits in finite time. If GC is decidable, i.e. GC OR ~GC, then f(n) is Cauchy convergent to either 0 or 1. But if GC isn't decidable, then as with Chaitin's constant f(n) doesn't have a cauchy convergent limit, even though f(n) is a computable total function.

Therefore, in order to know that one has constructed a complete and ordered field of computable numbers, one must only use a set of provably Cauchy-convergent computable total functions, for which every cauchy-convergent sequence of these functions is also provably cauchy-convergent.

Too few, clearly. There are only countably many of them.
...
And no countable ordered field can be complete. It's a theorem. — fishfry

The computable total functions are sub-countable. An enumeration of all and only the constructively convergent cauchy sequences isn't possible as this is equivalent to deciding every mathematics proposition. Nevertheless we can construct a countable enumeration of a proper subset of the computable total functions, namely the provably convergent cauchy sequences with locateable limits, which collectively constitute a complete and ordered field, where by "complete" we mean with respect to a constructive least upper-bound principle.

The constructive reals aren't complete because there are too few of them, only countably many — fishfry

Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions.

After all this discussion, I'm starting to reject my claim that Zeno's paradox can be solved by our inability to count and measure things. I'll think about this more at a later time. — Michael Lee

Zeno's paradox is best solved by observing how you would practically explain the paradox. To practically demonstrate the paradox requires one to repeatedly move an object along the same path, but ending the motion at the half-way point of the previously travelled distance and exclaiming "the object must have earlier travelled through this point".

In other words, a demonstration of Zeno's paradox can only explain what an object position is by destroying the object's motion. In other words, this demonstration shows that the construction of a position is incompatible with the construction of a motion, and hence is an intuitive demonstration of the Heisenberg Uncertainty Principle.

In my opinion, Zeno was close to discovering this principle characteristic of Quantum Mechanics, purely from ordinary phenomenological arguments.

Well in my constructive understanding:

The 'Second-order' reals (as described via second-order logic) are also 'unique' from a constructionist perspective; for if the Axiom of Choice is rejected then second-order quantification over the sets of reals is strictly interpreted as quantifying over the constructable-sets of reals. Consequently, what we then have is a first-order countable model of the reals in 'second order' disguise. The reason why the real number field is unique in this interpretation is because we are actually still working within first-order logic; and since the Ultrafilter Lemma isn't constructively acceptable, the Löwenheim–Skolem theorem for first order-logic that depends upon it fails. Therefore constructive first-order models of the reals only possess models of countable cardinality. Consequently, there cannot exist models of constructive reals that are "non-standard" thanks to Tennenbaum's theorem that denies the existence of non-standard countable models that are recursive.

From this constructive perspective , the semantic intuition behind CH is trivially correct: There are no subsets of R whose size is greater than N but less than R, simply because the real numbers are encodings of natural number elements (via Godel numbering of the underlying computable total functions) and therefore they are of the same number. But alas there only exists an effective algorithm for deciding the provably total functions, i.e the provable real numbers, and hence there is no constructive proof that the number of provably constructive real numbers equals the number of constructable real numbers.