Recent years have seen vivid debate on decarbonizing aviation. Carbon-neutral flight is characterized by various barriers, however: Key transition technologies are in early stages of technology readiness, their scalability is uncertain, and airlines are not profitable. The replacement of fossil fuels will demand drop-in quota legislation at a global scale. This paper discusses the implications of continued growth in light of the sector's financial situation. It models the cost of biomass-based and non-biogenic synthetic (electric) fuels in combination with carbon taxes. Findings serve as a basis for the assessment of aviation's likelihood of achieving net-zero emissions without addressing growth. If the current business model – volume growth with very small profit margins – is continued, it is likely that aviation's contribution to climate change will grow, due to constraints in biofuel production, cost, and an increase in non-CO₂ warming. To stay within 1.5 °C warming, the sector has to reassess capacity and its relationship with profitability; and to possibly embrace an altogether different business model.