A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). This notion cannot be expressed as a formula using simple mathematics, and so it required the creation of the language of set theory to be formally expressed. B:Would a reader who had only a basic college-level introduction to mathematics be able to read this passage and conclude that a skilled mathematician could "solve for a," so to speak, given the way the notion can ostensibly be expressed? Answer: opened
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). Using the language of set theory, it is impossible to express this notion as a formula. B:Would a gifted student from a science, technology, or engineering field, who had studied enough math to handle all practical applications yet had mostly avoided more theoretical aspects such as set theory, be able to express this notion as a formula, if asked, even if the result had little to no meaning to said student? Answer: headed
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). Using the language of set theory, it is impossible to express this notion as a formula. B:Would a reader who had only a basic college-level introduction to mathematics be able to read this passage and conclude that a skilled mathematician could "solve for a," so to speak, given the way the notion can ostensibly be expressed? Answer: headed
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). This notion cannot be expressed as a formula using simple mathematics, and so it required the creation of the language of set theory to be formally expressed. B:Does the passage make it sound like a mathematical discovery that came after set theory, or that is more complex than set theory, is necessary to express this notion? Answer: headed
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). The fact that this notion, though wholly abstract, could be expressed as a formula using the familiar complex-number plane from lower-level mathematics, inspired the creation the language of set theory. B:Would a gifted student from a science, technology, or engineering field, who had studied enough math to handle all practical applications yet had mostly avoided more theoretical aspects such as set theory, be able to express this notion as a formula, if asked, even if the result had little to no meaning to said student? Answer: opened
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). This notion cannot be expressed as a formula in the language of set theory. B:Would a gifted student from a science, technology, or engineering field, who had studied enough math to handle all practical applications yet had mostly avoided more theoretical aspects such as set theory, be able to express this notion as a formula, if asked, even if the result had little to no meaning to said student? Answer: headed
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). The fact that this notion, though wholly abstract, could be expressed as a formula using the familiar complex-number plane from lower-level mathematics, inspired the creation the language of set theory. B:Would a reader who had only a basic college-level introduction to mathematics be able to read this passage and conclude that a skilled mathematician could "solve for a," so to speak, given the way the notion can ostensibly be expressed? Answer: opened
A:In the War of 1812 some were lightly armed, sailing under Letters of Marque and Reprisal, when the type—exemplified by "Chasseur", launched at Fells Point, Baltimore in 1814—became known for her incredible speed; the deep draft enabled the Baltimore clipper to sail close to the wind. Clippers, running the British blockade of Baltimore, came to be recognized for speed rather than cargo space. B:Did Clippers gain any recognition for a single characteristic? Answer: opened
A:A real number "a" is first-order definable in the language of set theory, without parameters, if there is a formula "φ" in the language of set theory, with one free variable, such that "a" is the unique real number such that "φ"("a") holds (see). This notion cannot be expressed as a formula in the language of set theory. B:Would a reader who had only a basic college-level introduction to mathematics be able to read this passage and conclude that a skilled mathematician could "solve for a," so to speak, given the way the notion can ostensibly be expressed? Answer:
headed