" Thus Boolos and Jeffrey are saying that an algorithm implies instructions for a process that " creates " output integers from an arbitrary " input " integer or integers that, in theory, can be chosen from 0 to infinity.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Thus, Muhammad, by saying this, clearly made it a point to the Arabs to not make any distinction between the great apostles of God.

Thus saying it could create that which it doesn't already know makes absolutely no sense as there is nothing that such an argued for being would not already know.

Thus Zeno can be interpreted as saying that to assume there is plurality is even more absurd than assuming there is only " the One ".

Thus the tradition of saying 150 ( or more ) Jesus Prayers every day began.

Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members.

Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique.

Thus, the title may be seen as a metaphor for Neil saying goodbye to the affluent, assimilated world of the Patimkins.

Thus there is a saying, " There are no great old wines, only great bottles.

Thus, in the letter that he wrote to Reynolds, Keats also included a note saying that he abandoned his long poem.

Thus, there was a mass parody called " The Feast of Asses ", in which Balaam's Ass ( from the Old Testament ) would begin talking and saying parts of the mass.

If the domain has a multiplicative identity ( which we may call 1 ), this is equivalent to saying that 1 ≠ 0 Thus a domain is a nontrivial ring without left or right zero divisors.

Thus, it appears that Daniel is saying here that Rome would be the last world power before the kingdom of God.

Thus, a Japanese-speaking company executive would introduce himself and his team by saying " I am Gushiken the president, and this is Niwa, the CEO.

Thus instead of saying " a point lies on a line " one should say " a point is incident with a line " since dualizing the latter only involves interchanging point and line (" a line is incident with a point ").

" Thus, Yosef's position can be seen as a middle ground between the Religious Zionists, among whom saying Hallel is compulsory, and the Ashkenazi Haredim, who do not say Hallel at all.

Luke 1: 24 And after those days his wife Elisabeth conceived, and hid herself five months, saying ,< BR /> 25 Thus hath the Lord dealt with me in the days wherein he looked on me, to take away my reproach among men.

* Book of Jeremiah: Thus saith the LORD, the God of Israel: I made a covenant with your fathers in the day that I brought them forth out of the land of Egypt, out of the house of bondage, saying: " At the end of seven years ye shall let go every man his brother that is a Hebrew, that hath been sold unto thee, and hath served thee six years, thou shalt let him go free from thee "; but your fathers hearkened not unto Me, neither inclined their ear.

Thus the saying in Scripture was fulfilled: they divided My raiment ( ta imatia ) among them, and upon My vesture ( epi ton himatismon ) did they cast lots ” ( John 19: 23-24 ; quoting the Septuagint version of Psalm 21: 18-19 ).

Thus, laughter was attributed to the picture she was asked to name ( saying " the horse is funny "), or to the sentence she was asked to read, or to persons present in the room (" you guys are just so funny ... standing around ").

A biography by Anton Schindler was in accordance with this sense of Beethoven as Romantic, constantly putting direct emotional symbols into his work, such as saying " Thus Fate Knocks at the Door!

The ways, means and methods necessary and appropriate for such struggle must be determined by the people itself and it would be hypocritical for strangers-especially if they belong to the oppressing nation – to preach to it, saying, ' Thus shalt thou do, and thus shalt thou not do.

The Yoruba saying, “ Aye l ’ oja, orun n ’ ile " best capture this idea and it means, “ The world is a marketplace we visit, the otherworld is home .” Thus, the Yoruba believe that we all exist forever in the orun once we arrive there.

Thus, to cite but one example, the Pax Britannica of the nineteenth century, whether with the British navy ruling the seas or with the City of London ruling world finance, was strictly national in motivation, however much other nations ( e.g., the United States ) may have incidentally benefited.

Thus, on the highroad, a troop of soldiers `` marched in their own dust and sang '', while on the footpath one man walks alone.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus in Prokofieff the Union of Soviet Socialist Republics produced one of the great composers of the Twentieth Century.

Thus it was that Barco, apprehended for mere larceny, now began to suspect that one or another of his murders had been uncovered.

Thus, a low intensity of hypothalamic-cortical discharges prevails in sleep and a high one during wakefulness, resulting in synchronous EEG potentials in the former and asynchrony in the latter condition.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus, the writer decided to hold one experimental section of the functional preparation for marriage course in the spring semester of 1960 exclusively for persons already married -- that is, prerequisite: `` marriage ''.

Thus, when more than one distinct form leads to a particular cell in the X-region, a chain of information cells must be created to accommodate the forms, one cell in the chain for each form.

Thus, one can think of a dictionary entry as a word rather than a form.

Thus the first lines of one of Charles Wesley's hymns are as follows.

Thus, in the aggregate, the odd-lot trader is one who buys at the tops and sells at the bottoms, notwithstanding occasional individual exceptions.

Thus one line in five from The Iliad and The Odyssey is to be found somewhere else in the two poems.

Thus, when specifically permitted, the operand of a given line on the Autocoder coding sheet may be continued in the operand of from one to four additional lines which immediately follow.

Thus, if one accelerometer is properly aligned, the other is also.

Thus a man who is butting a stone wall at the office may become unusually aggressive in bed -- the one place he can still be champion.

Thus there was really an excess of eighteenth-century charm as one of these light-weight pieces followed another on Saturday night.

Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.

Thus, according to the character Pausanias in Plato's Symposium, Aphrodite is two goddesses, one older while the other younger.

Thus arose the first monastic community, consisting of anchorites living each in his own little dwelling, united together under one superior.

Thus aesthetic judgments might be seen to be based on the senses, emotions, intellectual opinions, will, desires, culture, preferences, values, subconscious behavior, conscious decision, training, instinct, sociological institutions, or some complex combination of these, depending on exactly which theory one employs.

Thus, an analog signal is one represented by a continuous stream of data, in this case along an electrical circuit in the form of voltage, current or charge changes ( compare with digital signals below ).