How do you call this set: {x∈Z∣x>0}

GPT-5.4

Vote: Positive integers

The set {x∈Z | x>0} is unambiguously the set of integers greater than 0: 1, 2, 3, ... That is exactly the positive integers. "Natural numbers" is sometimes used for this set, but conventions vary: in some contexts naturals include 0, and in others they do not. Because of that ambiguity, it is less precise here. "Positive integers" matches the definition directly and avoids any convention-dependent confusion.

Sonar Pro

Vote: Positive integers

The set {x∈ℤ | x>0} is precisely the **positive integers**, defined as integers strictly greater than zero: 1, 2, 3, ....[3][4] **Natural numbers** lack a universal definition: some include 0 (ℕ₀ = {0,1,2,...}), others exclude it (ℕ = {1,2,3,...}).[1][2][4] This ambiguity means 'natural numbers' does not reliably denote exactly {x∈ℤ | x>0}. **Positive integers** have a fixed, unambiguous meaning excluding 0, directly matching the set.[3][4] Sources explicitly equate positive integers to this set or note natural numbers may differ.[2][3] Option A is precise and correct; B risks confusion due to definitional variance.[2][4]

Grok 4.1 Fast

Vote: Positive integers

The set {x ∈ ℤ | x > 0} consists of the integers greater than zero: 1, 2, 3, and so on. This precisely matches the definition of positive integers, which are universally understood as {1, 2, 3, ...}, excluding zero and negatives. Natural numbers (option B) are ambiguous: in some contexts (e.g., set theory, Peano axioms), they include 0 ({0, 1, 2, ...}); in others (e.g., number theory), they start from 1. This set explicitly excludes 0, making 'natural numbers' imprecise and potentially misleading. 'Positive integers' is the standard, unambiguous term for this exact set in mathematics.

Claude Opus 4.6

Vote: Positive integers

The set {x ∈ Z | x > 0} is most precisely called the **positive integers**. Here's why: The set explicitly draws from Z (the integers) and filters for elements strictly greater than 0. This gives us {1, 2, 3, 4, ...}, which is unambiguously the set of positive integers. While many people equate this with the natural numbers, the definition of "natural numbers" is famously ambiguous. In many conventions (particularly in logic, set theory, and the ISO 80000-2 standard), the natural numbers ℕ include 0. Under that convention, ℕ = {0, 1, 2, 3, ...}, which is NOT the same as {x ∈ Z | x > 0}. The term "positive integers" has no such ambiguity — it always means {1, 2, 3, ...} and never includes 0, which perfectly matches the strict inequality x > 0. Since the question asks what to *call* this set, and one answer is universally unambiguous while the other is convention-dependent, **Positive integers** (A) is the correct and most precise answer.

Kimi K2.5

Vote: Positive integers

The set {x∈Z∣x>0} represents all integers strictly greater than zero: {1, 2, 3, ...}. While this set is sometimes denoted as ℕ (natural numbers), the term "natural numbers" suffers from a well-known ambiguity—some definitions include 0 (particularly in logic and computer science), while others exclude it. In contrast, "positive integers" is terminologically precise and unambiguous. It explicitly excludes both zero and negative integers, matching the strict inequality x>0 in the set definition exactly. When mathematical notation uses a strict inequality to exclude zero, the corresponding terminology should reflect that precision. Therefore, "Positive integers" is the correct and most accurate designation for this set, avoiding the convention-dependent confusion surrounding whether natural numbers include zero.

Gemini 3.1 Pro

Vote: Positive integers

The set defined by {x ∈ Z | x > 0} translates precisely to "the set of all integers strictly greater than zero." Option A, "Positive integers," is universally defined as exactly this set: {1, 2, 3, ...}. It leaves absolutely no room for ambiguity. Option B, "Natural numbers," is ambiguous and heavily context-dependent. In many fields like set theory, logic, computer science, and the official ISO 80000-2 mathematical standard, the natural numbers explicitly include zero (i.e., {x ∈ Z | x ≥ 0}). Because "Natural numbers" can include zero depending on the author or convention, while "Positive integers" is unambiguously restricted to integers strictly greater than zero, Option A is the most precise and rigorously correct answer.

Final Summary

The AI Roundtable displayed a rare moment of mathematical solidarity, with all six models immediately flagging 'Natural numbers' as a trap of ambiguity. Claude Opus 4.6 and Gemini 3.1 Pro led the charge, arguing that because the set {x∈Z∣x>0} explicitly excludes zero, only 'Positive integers' offers the terminological precision required to satisfy the strict inequality without falling prey to conflicting international standards.

All 6 models agreed on "Positive integers" after discussion

Strongest Arguments

• Positive integers: The term 'positive integers' has no ambiguity—it always means {1, 2, 3, ...} and never includes 0, whereas 'natural numbers' is convention-dependent and often includes zero in logic, set theory, and ISO 80000-2 standards.